Vision for QCD to the 2030s and Beyond111Based on the closing talk at the 28th
Workshop on Deep-Inelastic Scattering and Related Subjects, Virtual Event,
Stony Brook, April 12-16, 2021.

George Sterman

C.N. Yang Institute for Theoretical Physics and Department of Physics and
Astronomy

Stony Brook University, Stony Brook New York 11794-3840, U.S.A.
<EMAIL_ADDRESS>
## Abstract

In this closing talk of the DIS 2021 Workshop, I review some of the lessons
we’ve learned about quantum chromodynamics, and reflect on what we may hope to
learn in the coming years.

###### Contents

  1. 1 Introduction
  2. 2 QCD in the Grand Scheme of Things
  3. 3 Looking Closely, and from Afar
    1. 3.1 Inclusive and exclusive observables and factorization
    2. 3.2 From jet cross sections to long-time behavior
  4. 4 Putting the Pieces Together
  5. 5 Concluding Thoughts

##  1 Introduction

I have been asked to provide a vision in this talk, but it isn’t meant to be
an exercise in prediction. The year 2030 is still a long way off. The study of
quantum chromodynamics, however, is now driven as much by evolving
experimental capabilities as by the evolution of theory, so we have some sense
of what may happen in the meantime.

Ten years ago, the last of the great Standard Model discovery machines, the
Tevatron, closed up shop, following LEP and HERA, while RHIC entered its
second fruitful decade. The decade just past also saw the historic LHC Runs I
and II, as CEBAF transitioned from 6 to 12 GeV at Jefferson Lab. All these
milestones were reflected in the DIS series of conferences.

Starting with RHIC, many accelerator capabilities have been designed with QCD
in mind, at JLab of course, and in the decade unfolding, the EIC. The LHC was
not built for QCD, but the insightful designs of its detectors make it (of
necessity) a powerful QCD discovery tool.

Over the past twenty years, QCD has brought nuclear and particle physics
(back) together. Roads from Newport News and Upton lead to Geneva (and back).
The specifically QCD experimental capabilities that will link the 2020s and
the 2030s, including fixed target experiments at Fermilab, JLab, CERN and
Brookhaven, have already paved the way for the nascent Electron Ion Collider
project, based on the demonstrated need for high statistics to reveal the
structure of the nucleon, and high energy to unlock the dense gluonic matter
from which the mass of the visible universe is primarily generated. Those same
energy and statistics will provide the means to study the emergence of
hadronic from partonic matter.

The story that follows is one sketch of some lessons we have learned about the
role QCD in nature, and some we may hope to learn in the coming years. Most of
the topics I’ll touch on can be found, with many references, in the survey [1]
and recent dedicated study [2] of QCD physics at the Electron-Ion Collider.
The limited selection of references provided below reflects my own recent
reading and knowledge of work that illustrates the discussion that follows. It
is by no means meant to review all the important work that has been done in
these areas.

##  2 QCD in the Grand Scheme of Things

Very early in the history of the universe we know, quarks and gluons secluded
themselves to a nearly vanishingly small, and ever-decreasing, proportion of
space, occupying something like one $10^{-45}$th of the volume of the
observable universe. The reason, so far as we understand it, lies in the
nonabelian phase invariance of the quarks: three utterly indistinguishable
colors, connected by gluonic excitations. This, of course, is quantum
chromodynamics.

From inside the nucleon, quarks send and receive signals to and from the
outside world, using the rest of the Standard Model. Gluons are very busy, of
course, but they let the quarks and antiquarks do all the talking. Nucleons
and nuclei give electrons a reason to stick around and form the world we can
see. But the QCD degrees of freedom are always available, lurking in the
vacuum, ready to lend a hand and work alongside the rest of the Standard
Model, whenever enough energy arrives in the neighborhood.

Figure 1: Left: The electroweak process mediating the annihilation of lepton
pairs to hadrons. Right: The total cross section for electron-positron
annihilation to hadrons [3].

The very same theory possesses asymptotic freedom, which has opened windows to
its fundamental degrees of freedom. Just knowing that QCD is asymptotically
free is enough to get a good estimate of an important physical process, the
total $\rm e^{+}e^{-}$ annihilation cross section to hadrons, as if the theory
were free. The familiar electroweak process mediating the transition of
electron-positron pairs to quark pairs, and hence to the world of hadrons, is
shown on the left of Fig. 1. The inclusive cross section, on the right of Fig.
1, is very much the same as what it would have been if the quarks were able to
go their merry ways, without further interactions. At long distances, the same
theory forms only bound states, the mesons and baryons of the quark model,
and, as discovered in recent years, exotics of a surprisingly wide variety
[4]. A few of the quark model states are visible on the right of the figure.
Yet all of these “long-distance” phenomena do not radically change the
inclusive annihilation cross section.

##  3 Looking Closely, and from Afar

Much of our knowledge of QCD comes from a judicious choice of quantities to
observe. Inclusive cross sections offer a window into short-distance
processes, while providing a partonic map of the structure of hadrons.
Inclusive, of course, is a relative term, and the observation and analysis of
the more fine-grained structure of final states, particularly involving jets
of hadrons, affords a variable resolution into the long-time behavior of the
theory. This is the case perturbatively, and this approach may some day
provide a theory of the transition from partonic to hadronic descriptions of
strongly-interacting matter.

###  3.1 Inclusive and exclusive observables and factorization

In a broad sense, the “universal” form of hard-scattering observables, both
cross sections and amplitudes, is a factorization, very schematically of the
form [5, 6]

$S\ =\ C\ \times\ F\,.$ (1)

The left-hand side is an observable quantity, while the right-hand side
separates a calculable “short distance” factor, represented by $C$, which can
be computed in perturbation theory, from a “long distance” quantity, $F$
accessible only to experimental determination, and increasingly, to numerical
simulation. The product represents in general a convolution, most simply in
partonic momentum fraction.

The classic example of Eq. (1) is, of course, deep-inelastic scattering,
illustrated by Figs. 2 and 3. Figure 2 illustrates a typical DIS event. In the
figure, the diagonal axes represent the light cone, around which the extended
proton, of momentum $p$ and represented as a cylinder, arrives. A quark of
flavor $f$ and fractional momentum $x$ is suddenly scattered by the exchange
of a photon of momentum $q$. The full final state is generally complex, and
involves many other particles emerging from the fractured proton.

Despite this complexity, we can measure the momentum transfer simply by
observing the final-state electron. From its final-state momentum,
$k^{\prime}$ in the figure, we can determine the momentum transfer $q$. A
large, spacelike momentum transfer $q$ localizes the hard scattering to points
that differ in the amplitude and the complex conjugate amplitude by a nearly
lightlike distance. This provides a clean separation between initial and final
states. In fact, we can do even better by a judicious observation of a
particle or jet in the direction of the scattered quark’s momentum,
$p^{\prime}=xp+q$ in the figure. In this case, the difference betweent the
scattering event in the amplitude and complex conjugate remains near the
lightcone, but deviates slightly in the transverse plane. Appropriate sums
over final states provide measurements of parton correlations in the nucleon,
including spin.

Figure 2: A typical event in deep-inelastic scattering.

In all of these cases, fully or semi inclusive, a sum combines the total
probabilities of specific final states. The total result must add up to unity,
even if we cannot calculate the probability for any final state individually.
Corrections arise only from distances of order $1/\sqrt{q^{2}}$, and can hence
be computed perturbatively, Long-distance contributions from final states,
which are infrared divergent in perturbation theory, cancel. The result is the
“coefficient” function $C$ in Eq. (1).

The amplitudes for the noncancelling initial states can be interpreted as a
parton distribution [7], illustrated in Fig. 3. Here, a proton comes in from
the past to some fixed point in spacetime, where a quark is removed from it. A
quark of the same flavor appears at a nearly (but in general not quite)
lightlike separation. The parton distribution is the amplitude for this proton
to reemerge and travel undisturbed into the future. As suggested in Fig. 3, if
the spin carried by the quark that disappears is correlated with the spin of
the one that appears, we have a polarization-dependent distribution. All of
these parton distributions can be written as field-theoretic matrix elements.
For fully inclusive DIS, we need the“collinear” parton distributions

$F_{Q}(x)\ =\ \int\frac{d\lambda}{2\pi}\,e^{-i\lambda\,xP\cdot n}\langle
P|\,\bar{Q}(\lambda n)\,\Gamma\,Q(0)\,|P\rangle\,,$ (2)

where $Q(x)$ represents a quark (or gluon) field, and where the vector
$n^{\mu}$ is the light cone direction just opposite to the proton in the
center of mass. Here $\Gamma$ represents a Dirac or vector projecction and,
for the experts, an appropriate gauge link. When transverse momenta are
observed in the final state, we translate the field $\bar{Q}$ off the light
cone as in the figure, and Fourier transform in that “impact parameter”
distance, giving a transverse-momentum dependent, or “TMD” distribution. The
factorization of Eq. (1) is then in transverse as well as longitudinal,
degrees of freedom [9, 8].

Figure 3: Illustration of the matrix element for a collinear or transverse
momentuum distribution. A quark is removed from the proton, and reinserted at
a point on or near the opposite-moving light cone.

Factorizations like Eq. (1) with parton distributions of the type in Eq. (2)
are only the first term in a series expansion in momentum transfer. So-called
“higher twist” terms involve more fields, and depend on matrix elements such
as [10, 11] $\langle P|\bar{Q}(x)\Gamma G(y)Q(0)|P\rangle$, with $Q$ a quark
and $G$ a gluon field. For some observables, especially involving spin, this
is the leading effect, offering information on short-distance correlations
between partons of different flavor.

As broad as the forms of Eq. (1) are for inclusive and semi-inclusive cross
sections, these are only one route from experiment to the QCD matrix elements
that reveal nucleon structure. With the matrix element interpretation of
parton distributions as inspiration, we can find similar factorized
expressions for exclusive amplitudes. The classic example is the process of
deeply-virtual Compton scattering, illustrated by Fig. 4. Here we can study a
unique quantum-mechanical amplitude, corresponding to the process in the
picture.

Figure 4: The process encoded in a generalized parton distribution.

In this reaction, a proton and an electron collide, and the electron exchanges
an off-shell photon with a quark. The quark then travels for some distance in
the proton, after which it emits an on shell photon. The proton then reforms
itself, with the quark starting out in a new place and with a slightly
different momentum. This prccess is described by matrix elements of the
general form, [12, 13, 14]

$M\left(x,z,\Delta^{2}\right)\ =\ P\cdot
n\,\int\frac{d\lambda}{2\pi}\,e^{i\lambda xP\cdot n}\langle
P+q|\,\bar{Q}(-\lambda n)\,\Gamma\,Q(\lambda n)\,|P\rangle\,,$ (3)

for a quark field $Q$, in which a relatively small net momentum transfer, $q$
redirects the proton. This matrix element, corresponding to the figure,
depends on the longitudinal momentum $zp\cdot n$ that the quark loses in the
process, along with a corresponding transverse momentum transfer $\bf\Delta$.
It is from matrix elements such as these that the angular momentum and similar
non-local information on hadronic structure can be formulated in terms of
partonic degrees of freedom.

Taken together with the scattered electron, this process corresponds
experimentally to a very distinctive final state, with a slightly deflected
nucleon accompanied by an on-shell photon. The new sets of nucleon
correlations made available in these experiments are beginning to reveal the
stationary nucleon state in the language of partonic degrees of freedom. Such
quantities are also increasingly accessible to lattice calculations [15, 16].
These are among the most exciting contemporary confrontations of theoretical
with experimental physics of the strong interactions.

###  3.2 From jet cross sections to long-time behavior

The experimental studies described above all depend on grouping together
events with similar final states. For the total DIS cross section, these are
all states that share a scattered electron with the same momentum transfer and
energy loss. For studies of generalized parton distributions, these are the
much smaller, more exclusive class of final states with a scattered electron,
a high energy photon and a proton. Another window into short-distance
phenomena is provided by sets of states with similar global flows of energy.
These states, characterized by collimated sets of high energy particles, are
called jet states [17, 18]. Jet final states also offer a direct picture of
processes that take place at the elementary level, the scattering of quarks,
the creation of quark pairs, the decay of heavy particles, and the radiation
of gluons.

The analysis of jet events has taken on a new life, as we recognize that they
should be regarded as a source of‘ ‘big data”, and hence as a target for “deep
learning” [19, 20]. Looking at a detector display, our natural intellegence
readily identifies the jets qualitatively, relying on charged particle tracks
and such symbolic translations of energy scale as to the length or color of
bars representing particles or calorimeter signals. The information in
ensembles of jets, each containing perhaps dozens of particles of various
energies and flavor content, is the outcome of the complex process we
understand as perturbative showering with nonperturbative corrections,
culminating in hadronization [21, 22, 23, 24, 25]. For semi-inclusive or
inclusive but “weighted” cross sections, resummed perturbation theory, often
in the language of effective theories, [26, 27, 28, 29] supplemented by power
corrections [30, 31] can describe a wealth of data.

The flow of energy itself has a field theoretic analog, and much current
analysis revolves around exploring the properties of operators involving the
energy-momentum tensor [32, 33, 34],

$\displaystyle{\cal E}\left(\hat{n}\right)\ =\
\lim_{R\rightarrow\infty}R^{2}\,\int_{0}^{\infty}dt\,\hat{n}_{i}\,T_{0i}\left(t,R\hat{n}\right)\,,$
(4)

which measures energy flowing in the specific direction $\hat{n}$, as seen by
a detector. Correlations between particle jets can be generated from
expectations of products of these operators in “initial” states $|I\rangle$,
such as

$C\left(E(\hat{n}_{1})\dots E(\hat{n}_{a}),x\right)\ =\ \langle
I|j_{\mu}(x)\prod_{i=1}^{a}{\cal
E}\left(\hat{n}_{i})\right)j^{\mu}(0)|I\rangle\,,$ (5)

with $j^{\mu}$ an electroweak current. Such correlations can be calculated
perturbatively. Integrating over angular regions, we can measure energy
correlations over differing angular separations [35].

The dependence of angular correlations in energy flow on the medium traversed
by partons on their way from a hard scattering to the final state has been,
and remains, a sensitive probe of the properties of hot and cold nuclear media
[36, 37]. Varying energies, reactions, flavors and luminosities will be
necessary to explore fully these fundamental properties of matter. Interest is
also increasing on “hybrid” expectation values, involving both the energy-
momentum tensor and other conserved charges [38, 39]. The exploration of these
correlations will hopefully lead to new insights on the final stages of the
translation from the language of partons into that of hadrons.

##  4 Putting the Pieces Together

What might we expect from theory in the coming years? Here is a personal
perspective, very much ready to be supplemented by novel ideas.

We understand QCD best at its endpoints. Its dynamics at the very moment of a
hard scattering is well-described in perturbation theory, which we can use to
evolve to length scales approaching $1/\Lambda_{\rm QCD}$. At the longest
scales, the quark model and effective theories based on its symmetries also
afford quantitative descriptions. In between, perhaps we can think of a
process by which current quarks generate mass from radiative energy, becoming
constituent quarks. How to quantify this transition? Are there observables
sensitive to this period? Photons radiated in this stage might be an example
[40, 41].

Final states provide a countable amount of information, and we should be ready
to count it all, to the extent possible. The detailed census within each jet
encodes a set of stories. Jet substructure analyses are beginning the process
of breaking this code, but it will surely take many new insights, ideas, and
computing power [42, 43, 44].

A judicious use of machine learning, new ideas of event display and perhaps
quantum information will lead to the demystification of the transition between
the language of partons and that of hadrons. For amplitudes and inclusive
cross sections, we’ll continuously reevaluate to what orders can we calculate.
Analytic fixed-order progress will continue, hand-in-hand with mathematical
insight [45, 46, 47, 48, 49], while numerically, higher orders will become
more accessible for cross sections, and experimental cuts easier to implement.

We’ll hopefully see algorithmic evaluations of complex QCD amplitudes and
cross sections [50]. Why might this be possible? It’s the magic of unitarity
[51, 52, 53], the same conservation of probability that enables us to
factorize parton distributions in DIS and makes jet cross sections calculable.
We will then be able to avoid infrared divergences before we integrate over
phase space and impose experimental cuts. We will learn to calculate in four
dimensions [54, 55] and limit perturbation theory to finite times. This may
help make room for a new theory of hadronization.

QCD will be more and more embedded in the Standard Model, with flavor issues
coming to the fore at all scales, from $g-2$ all the way to future super high-
energy colliders. Also with the help of the lattice, we will learn more of the
correlations within hadrons and in exotic states of matter in the lab and in
the universe. Higher twist will be subsumed into a theory of quark-hadron
duality [56], building on the kind of analysis that led to QCD sum rules,
relating hadronic properties to the operator product expansion. A solution to
the strong CP may arise in the coming decade, perhaps in connection with an
observation of dark matter. Perhaps what we learn about QCD will suggest other
scenarios for early stages of the universe and matter under extraordinary
conditions. This would bring us back to the beginning.

##  5 Concluding Thoughts

In many ways, quantum chromodynamics is the exemplary quantum field theory. It
exhibits highly nontrivial phase structure, accessible now in ion collisions,
and perhaps through its collective flow in smaller systems too. Its classical
solutions, the instantons, opened new perspectives in geometry, and its
perturbative amplitudes have echos in the mathematics of complex analysis and
both classical and quantum general relativity. It serves as a benchmark for
more manageable theories, especially conformal quantum field theories and
related bootstrap programs.

Even more generally, any answer to the question “what is quantum field
theory?” must encompass and find inspiration from quantum chromodynamics. The
quantum field theory of our day is a bit like the calculus of Newton and
Leibnitz, not yet fully defined, but extraordinarily effective. This is one
reason why QCD continues to be guided by experiment. For example, only through
experience could we know that the particle jets of perturbative QCD would be
there at high energy in the laboratory.

Quantum chromodynamics is approaching its fiftieth year, but for a fundamental
theory, it is still young. Newtonian gravity, for example, is still going
strong at 334 since it appeared in the Principia. Quantum chromodynamics
itself remains an astounding discovery, with its mysterious unbroken color
symmetry. Even more, QCD makes it possible for us to face the challenge of
bridging the fundamental and the emergent phenomena of the natural world. This
is a hallmark of 21st Century science, and indeed of human thought.

Finally, at the time of this writing, we mark with sadness the loss of Steven
Weinberg, whose work and thought echo through these short pages and far
beyond, guiding how we think of quantum chromodynamics and so much else in
contemporary theoretical physics. His was a paragon of a life in science.

## Acknowledgements

Congratulations to the organizers of DIS XXVIII: 2020 / 2021, for making it
happen, and to all who took part, with the hope and expectation that DIS XXIX
will reflect a regained, and perhaps reimagined, freedom.

#### Funding information

This work was supported in part by the National Science Foundation, through
award PHY-1915903.

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# Gradient estimates for the Lagrangian mean curvature equation with critical
and supercritical phase

Arunima Bhattacharya  Department of Mathematics
University of Washington, Seattle, WA, U.S.A<EMAIL_ADDRESS>,  Connor
Mooney  Department of Mathematics
University of California, Irvine, CA, U.S.A<EMAIL_ADDRESS>and  Ravi
Shankar  Department of Mathematics
Princeton University, Princeton, NJ, U.S.A<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove interior gradient estimates for the Lagrangian mean
curvature equation, if the Lagrangian phase is critical and supercritical and
$C^{2}$. Combined with the a priori interior Hessian estimates proved in
[Bha21, Bha22], this solves the Dirichlet boundary value problem for the
critical and supercritical Lagrangian mean curvature equation with $C^{0}$
boundary data. We also provide a uniform gradient estimate for lower
regularity phases that satisfy certain additional hypotheses.

##  1\.  Introduction

In this paper, we study a priori interior gradient estimates in all dimensions
for the Lagrangian mean curvature equation

(1.1) $F(D^{2}u)=\sum_{i=1}^{n}\arctan\lambda_{i}=\psi(x),\qquad x\in
B_{1}(0)\subset\mathbb{R}^{n},$

under the assumption that $|\psi|\geq(n-2)\frac{\pi}{2}$. Here,
$u:B_{1}\to\mathbb{R}$ has gradient $Du$ and Hessian matrix $D^{2}u$, with
eigenvalues $\lambda_{i}$. We will denote $B_{r}=B_{r}(0)$ throughout.

When the phase $\psi$ is constant, denoted by $c$, $u$ solves the special
Lagrangian equation

(1.2) $\sum_{i=1}^{n}\arctan\lambda_{i}=c,$

or equivalently,

$\cos c\sum_{1\leq 2k+1\leq n}(-1)^{k}\sigma_{2k+1}-\sin c\sum_{0\leq 2k\leq
n}(-1)^{k}\sigma_{2k}=0.$

Equation (1.2) originates in the special Lagrangian geometry by Harvey-Lawson
[HL82]. The Lagrangian graph
$(x,Du(x))\subset\mathbb{R}^{n}\times\mathbb{R}^{n}$ is called special when
the argument of the complex number $(1+i\lambda_{1})...(1+i\lambda_{n})$, or
the phase $\psi$, is constant, and it is special if and only if $(x,Du(x))$ is
a (volume minimizing) minimal surface in
$(\mathbb{R}^{n}\times\mathbb{R}^{n},dx^{2}+dy^{2})$ [HL82].

More generally, for (1.1), it was shown in [HL82, (2.19)] that the mean
curvature vector $\vec{H}$ of the Lagrangian graph $(x,Du(x))$ is
$J\nabla_{g}\psi$, where $\nabla_{g}$ is the gradient, and $J$ is the almost
complex structure on $\mathbb{R}^{n}\times\mathbb{R}^{n}$. Note that
$|\vec{H}|_{g}$ is bounded for $\psi\in C^{1}$. In the complex setting, a
local version of the deformed Hermitian-Yang-Mills equation for a holomorphic
line bundle over a compact Kähler manifold is represented by equation (1.1).

The notions of critical and supercritical phases were introduced by Yuan
[Yua06]. The Lagrangian angle $\theta(\lambda)=\sum_{i}\arctan\lambda_{i}$ is
critical if $|\theta|=(n-2)\pi/2$ and supercritical if $|\theta|>(n-2)\pi/2$.
We recall that the variable phase $\psi(x)$ is called critical and
supercritical if $|\psi(x)|\geq(n-2)\pi/2$, and supercritical if
$|\psi(x)|\geq(n-2)\pi/2+\delta$ for some $\delta>0$. It was shown in [Yua06,
Lemma 2.1] that the level sets $\\{\lambda:\theta=c\\}$ are convex for
critical and supercritical phases. In particular, there are Evans
[Eva82]-Krylov [Kry83]-Safonov[Saf84, Saf89] $C^{2,\alpha}$ estimates if
$D^{2}u$ is bounded, and $\psi(x)$ is Hölder continuous.

In this paper, for $C^{2}$ critical and supercritical phases, we solve the
Dirichlet problem for $C^{0}$ boundary data by establishing the missing
interior gradient estimates. Interior Hessian estimates for supercritical
$C^{1,1}$ phases were shown by Bhattacharya in [Bha21, Theorem 1.1]; interior
Hessian estimates for critical and supercritical phases follow verbatim from
the calculations done in [Bha21] (see [Bha22, Remark 2.1]); interior gradient
estimates for supercritical $C^{1}$ phases were derived in [Bha21, Theorem
1.2].

Our main result is a gradient estimate for arbitrary $C^{2}$ critical and
supercritical phases.

######  Theorem 1.1.

Let $u$ be a $C^{3}(\overline{B_{1}})$ solution of (1.1) on
$B_{1}(0)\subset\mathbb{R}^{n}$, where $\psi\in C^{2}(B_{1})$ satisfies
$\psi\geq(n-2)\frac{\pi}{2}$. Then

(1.3) $|Du(0)|\leq
C(n,||D^{2}\psi||_{L^{\infty}(B_{1})})\left(1+(\text{osc}_{B_{1}}u)^{2}\right).$

We state the following Hessian estimate combining [Bha21, Theorem 1.1] and
[Bha22, Remark 2.1].

######  Theorem 1.2.

Let $u$ be a $C^{4}$ solution of (1.1) on $B_{R}(0)\subset\mathbb{R}^{n}$,
where $\psi\in C^{2}(B_{R})$, and $\psi\geq(n-2)\frac{\pi}{2}$. Then we have

(1.4) $|D^{2}u(0)|\leq C\exp[C\max_{B_{R}(0)}|Du|^{2n-2}/R^{2n-2}]$

where $C$ is a positive constant depending on $||\psi||_{C^{2}(B_{R})}$ and
$n$.

As an application, we solve the following Dirichlet boundary value problem
with $C^{0}$ boundary data.

######  Corollary 1.1.

Suppose that $\phi\in C^{0}(\partial\Omega)$ and
$\psi:\overline{\Omega}\rightarrow[(n-2)\frac{\pi}{2},n\frac{\pi}{2})$ is in
$C^{2}(\overline{\Omega})$, where $\Omega$ is a uniformly convex, bounded
domain in $\mathbb{R}^{n}$. Then there exists a unique solution $u\in
C^{3}(\Omega)\cap C^{0}(\overline{\Omega})$ to the Dirichlet problem

(1.5)
$\displaystyle\begin{cases}F(D^{2}u)=\sum_{i=1}^{n}\arctan\lambda_{i}=\psi(x)\text{
in }\Omega\\\ u=\phi\text{ on }\partial\Omega\end{cases}$

The solution $u$ is, in fact, in $C^{3,\alpha}(\Omega)$ for any
$\alpha\in(0,\,1)$, by classical uniformly elliptic theory.

The Dirichlet problem for a broad class of fully nonlinear, elliptic equations
of the form $F(\lambda[D^{2}u])=f(x)$ was first studied by Caffarelli-
Nirenberg-Spruck in [CNS85], where they proved the existence of classical
solutions under various hypotheses on the function $F$ and the domain
$\Omega$. In [HL09], Harvey-Lawson studied the Dirichlet problem for fully
nonlinear, degenerate elliptic equations of the form $F(D^{2}u)=0$ on a smooth
bounded domain in $\mathbb{R}^{n}$. The existence and uniqueness of continuous
viscosity solutions to the Dirichlet problem for (1.2) with continuous
boundary data was shown in [HL09, Yua08]; see also [Bha20]. In [BW10],
Brendle-Warren studied a second boundary value problem for the special
Lagrangian equation.

For subcritical phases $|c|<(n-2)\pi/2$, interior regularity is not
understood. For critical $|c|=(n-2)\pi/2$ and supercritical $|c|>(n-2)\pi/2$
phases, interior gradient estimates were established by Warren-Yuan [WY09a,
WY10], and also Yuan’s unpublished notes from 2015. Interior Hessian estimates
for dimension $n=2$ were shown by Heinz, for $|c|=\pi/2$ in dimension $n=3$ by
Warren-Yuan [WY09b], and for general dimension $|c|\geq(n-2)\pi/2$ by Wang-
Yuan [WY14]; see also Li [Li19] for a compactness approach and Zhou [Zho22]
for estimates requiring Hessian constraints which generalize criticality.
Because the level set of the PDE is convex for critical and supercritical
phases, the Evans-Krylov theory yields interior analyticity. The singular
$C^{1,\alpha}$ subcritical phase solutions by Nadirashvili-Vlăduţ [NV10] and
Wang-Yuan [WY13] show that interior regularity is not possible for subcritical
phases, without an additional convexity condition, as in Bao-Chen [BC03],
Chen-Warren-Yuan [CWY09], and Chen-Shankar-Yuan [CSY22], and that the
Dirichlet problem is not classically solvable for arbitrary smooth boundary
data. Interior gradient estimates for continuous boundary data are widely
open. Global gradient estimates requiring Lipschitz boundary data were shown
by [Lu22]. Homogeneous viscosity solutions of degree less than two were shown
to not exist by Nadirashvili-Yuan [NY06]. The non-existence result of Mooney
[Moo22] shows that counterexamples for interior $C^{1}$ regularity may be
difficult to construct.

If the Lagrangian angle is not necessarily constant, then less is understood.
In [HL19], Harvey-Lawson introduced a condition called “tameness” on the
operator $F$, which is a little stronger than strict ellipticity and allows
one to prove comparison. In [HL21], tamability was established for the
supercritical Lagrangian mean curvature equation. In [CP21], Cirant-Payne
established comparison principles for the Lagrangian mean curvature equation
provided the Lagrangian phase is restricted to the intervals
$((n-2k)\frac{\pi}{2},(n-2(k-1))\frac{\pi}{2})$ where $1\leq k\leq n$, which
in turn solves the Dirichlet problem on these intervals as shown in [HL21,
Theorem 6.2]. Hessian estimates for convex smooth solutions with $C^{1,1}$
phase $\psi=\psi(x)$ were obtained by Warren [War08, Theorem 8]. For convex
viscosity solutions, interior regularity was established for $C^{2}$ phases;
see Bhattacharya-Shankar [BS20b, BS20a]. For supercritical phases
$|\psi(x)|\geq(n-2)\pi/2+\delta$, there is a comparison principle, and the
Dirichlet problem was solved in Collins-Picard-Wu [CPW17], Dinew-Do-Tô
[DDT18], Bhattacharya [Bha20], and interior gradient estimates were
established in [Bha21]. Interior Hessian estimates for supercritical phases
were established in [Bha21]. Interior Hessian estimates for critical and
supercritical phases $|\psi(x)|\geq(n-2)\frac{\pi}{2}$, follow verbatim from
the calculations done in [Bha21] (see [Bha22, Remark 2.1]): The proof of the
Hessian estimate in [Bha21, Theorem 1.1] does not require a negative lower
bound on the lowest eigenvalue. For supercritical phases in dimension $n=2$, a
simplified proof [Bha22] was given for interior Hessian estimates using the
super-isoperimetric inequality of Warren-Yuan [WY09a], avoiding the Michael-
Simon mean value inequality [MS73]. In the case that $\phi$ is Lipschitz,
Corollary 1.1 can be obtained by proving a global gradient estimate, as in
[Lu22]. The existence of interior gradient estimates for the challenging
borderline case of critical and supercritical phase has until now remained
open. In this paper, we successfully solve this problem for $C^{2}$ phases.

Our approach to prove interior gradient estimate Theorem 1.1 accounts for the
smallness of the gradient of the phase near its minimizing, critical values,
using a pointwise interpolation inequality [NT70, Equation (3.11), pg. 19],
see also [Hor83, Lemma 7.7.2], valid for $C^{2}$ phases. For constant phases,
the gradient estimate is established using a maximum principle inequality. The
variable phase contribution to the inequality is a “bad term” depending on the
phase’s gradient. Although the PDE’s ellipticity degenerates at the critical
phase, making the bad term large, the smallness of the gradient at such points
provides a balance. Our proof, more generally, shows that an interior gradient
estimate holds when $\psi$ satisfies a certain first order differential
inequality; see Remark 2.2. Such an inequality is valid when the phase is any
of $C^{2}$, semi-concave, concave, or a supersolution of the infinity-Laplace
equation; see Remark 2.3.

On the other hand, the gradient vanishes at slower rates for $C^{1,\alpha}$
phases, and does not appear to balance the degeneration of the ellipticity in
our proof. But we note that certain Hölder continuous phases allow for
gradient estimates; see Remark 2.4. In such cases, the phase separates from
the critical value at such a large speed that the solution is nearly semi-
convex, as in the supercritical case.

Acknowledgements. A. Bhattacharya is grateful to Yu Yuan for bringing
attention to this problem and for many helpful discussions. R. Shankar thanks
Yu Yuan for helpful suggestions about the paper. A. Bhattacharya would like to
thank Pengfei Guan for pointing out that the interpolation inequality appears
in [NT70], as well as Tristan Collins for helpful conversations. C. Mooney
gratefully acknowledges the support of NSF grant DMS-1854788, NSF CAREER grant
DMS-2143668, an Alfred P. Sloan Research Fellowship, and a UC Irvine
Chancellor’s Fellowship.

##  2\. Proof of the gradient estimate

We modify the pointwise proof of [WY09a] and [Bha21] to bridge the constant
critical phase estimate of [WY09a] with the supercritical estimate of [Bha21].
The difference comes from how to treat the bad term involving $D\psi$.

Notation: we denote $a\sim b$ if $ca\leq b\leq Ca$, and $a\lesssim b$ if
$a\leq Cb$. Here, $c$ and $C$ are positive constants depending on $n$. We will
denote $a\lesssim_{\psi}b$ if the constant also depends on
$\|D^{2}\psi\|_{L^{\infty}(B_{1})}$. We denote $a\ll_{n}1$ if we are choosing
a small fixed constant $a$ depending on $n$. We assume summation under
repeated indices unless otherwise indicated.

Let $M=\text{osc}_{B_{1}}u>0$; replacing $u$ with $u-\min_{B_{1}}u+M$, we
assume that

(2.1) $\displaystyle M\leq u\leq 2M\qquad\text{in }B_{1}.$

Let $w=\eta|Du|+Au^{2}/2$, where $\eta=1-|x|^{2}$, and $A=3\sqrt{n}/M$. Let
$x_{0}\in B_{1}$ be where $w$ is maximized. After a rotation, we assume that
$D^{2}u$ is diagonal, with $u_{ii}=\lambda_{i}$. Let us assume that
$u_{n}\geq|Du|/\sqrt{n}>0$. Then for each $k$, at the max point $x_{0}$,

(2.2) $\displaystyle
0=\partial_{k}w(x_{0})=\eta\frac{u_{k}\lambda_{k}}{|Du|}+\eta_{k}|Du|+Auu_{k}.$

Since $A=3\sqrt{n}/M$ is sufficiently large, it follows that

(2.3) $\displaystyle\eta\lambda_{n}\frac{u_{n}}{|Du|}\in-(c(n),C(n))|Du|.$

It follows that $\lambda_{n}<0$ and

(2.4) $\displaystyle\eta\sim\frac{|Du|}{|\lambda_{n}|}.$

Since $|Du|\lesssim\eta|\lambda_{n}|$, we may assume that $|\lambda_{n}|>1$,
since otherwise the estimate is done. Moreover, as shown in [WY14, Lemma 2.5],
we know that $\lambda_{k}\geq|\lambda_{n}|$ for $k<n$ follows from
$\psi\geq(n-2)\pi/2$.

We now proceed to the second derivatives of $w$. Let $g=I+(D^{2}u)^{2}$ be the
induced metric $dx^{2}+dy^{2}$ on $(x,Du(x))$, with $g^{-1}=(g^{ij})$ its
inverse, and $g^{ij}=(1+\lambda_{i}^{2})^{-1}\delta_{ij}$ at $x_{0}$. Then at
$x_{0}$,

(2.5) $\displaystyle 0$ $\displaystyle\geq g^{ij}\partial_{ij}w(x_{0})$
$\displaystyle=\underbrace{g^{ij}\eta_{ij}|Du|}_{(C1)}+\underbrace{2g^{ij}\eta_{i}|Du|_{j}}_{(C2)}+\underbrace{\eta
g^{ij}|Du|_{ij}}_{(B)}+\underbrace{Aug^{ij}u_{ij}}_{(C3)}+\underbrace{Ag^{ij}u_{i}u_{j}}_{(G)}.$

The good term (G) absorbs the constant-phase terms (C1), (C2), and (C3) in the
constant phase and supercritical cases, as in [WY09a] and [Bha21]. The bad
term (B) contains variable phase contributions and will require closer
examination.

The good term (G):

(2.6) $\displaystyle Ag^{ij}u_{i}u_{j}\geq
A\frac{u_{n}^{2}}{1+\lambda_{n}^{2}}\sim
A\frac{|Du|^{2}}{\lambda_{n}^{2}}\sim\frac{A\eta|Du|}{|\lambda_{n}|}.$

The first constant phase term (C1):

(2.7) $\displaystyle
g^{ij}\eta_{ij}|Du|=-2\sum_{i}\frac{1}{1+\lambda_{i}^{2}}|Du|\gtrsim-\frac{1}{\lambda_{n}^{2}}|Du|\sim-\frac{\eta}{|\lambda_{n}|}.$

The second constant phase term (C2):

(2.8) $\displaystyle 2g^{ij}\eta_{i}|Du|_{j}\stackrel{{\scriptstyle
x_{0}}}{{\gtrsim}}-\sum_{i}\frac{1}{1+\lambda_{i}^{2}}\frac{u_{i}\lambda_{i}}{|Du|}\gtrsim-\frac{1}{|\lambda_{n}|}.$

The third constant phase term (C3):

(2.9) $\displaystyle
Aug^{ij}u_{ij}=Au\sum_{i}\frac{\lambda_{i}}{1+\lambda_{i}^{2}}\gtrsim-\frac{1}{|\lambda_{n}|}.$

The bad term (B), using third derivative calculation [Bha21, (2.4)]:

(2.10) $\displaystyle\eta g^{ij}|Du|_{ij}$ $\displaystyle=\eta
g^{ij}\frac{u_{ijk}u_{k}}{|Du|}+\eta\sum_{i}g^{ii}\frac{(|Du|^{2}-u_{i}^{2})\lambda_{i}^{2}}{|Du|^{3}}$
$\displaystyle\geq\eta\psi_{k}\frac{u_{k}}{|Du|}\gtrsim-\eta|D\psi|.$

We thus need to bound this inequality at $x_{0}$:

(2.11) $\displaystyle\eta|Du|\leq C(n)M(1+\eta\,|D\psi|\,|\lambda_{n}|).$

Letting $\phi=\psi-(n-2)\pi/2\geq 0$, we apply the pointwise interpolation
inequality for nonnegative $C^{2}$ functions in [Hor83, Lemma 7.7.2] on
$B_{\delta}(x_{0})$, where $\delta=1-|x_{0}|$;

(2.12) $\displaystyle|D\phi(x_{0})|^{2}$
$\displaystyle\leq\frac{\phi(x_{0})^{2}}{(1-|x_{0}|)^{2}}+2\|D^{2}\phi\|_{L^{\infty}(B_{1})}\phi(x_{0})$
$\displaystyle\lesssim_{\psi}\frac{\phi(x_{0})^{2}}{\eta^{2}}+\phi(x_{0}),$

where $\|D^{2}\phi\|_{L^{\infty}(B_{1})}$ denotes the maximum of the absolute
values of the eigenvalues of $D^{2}\phi$. Let us now recall the following
algebraic inequality, valid for $\lambda_{n}<0$ and $\lambda_{k}>0$ for $k<n$:

(2.13) $\displaystyle\psi$
$\displaystyle=(n-1)\frac{\pi}{2}-\sum_{i<n}\arctan(\frac{1}{\lambda_{i}})-\frac{\pi}{2}+\arctan(-\frac{1}{\lambda_{n}})$
$\displaystyle\leq(n-2)\frac{\pi}{2}+\frac{1}{|\lambda_{n}|}.$

Substituting this information into (2.11), combined with (2.4), yields

(2.14) $\displaystyle M^{-1}\eta|Du|$
$\displaystyle\lesssim_{\psi}1+\eta|\lambda_{n}|^{1/2}\lesssim
1+(\eta|Du|)^{1/2}.$

It follows that

(2.15) $\displaystyle\eta|Du|\lesssim_{\psi}M+M^{2}\lesssim
C(n,\|\psi\|_{C^{2}(B_{1.5})})(1+(\text{osc}_{B_{1}}u)^{2}).$

The $Au^{2}/2$ term in $w(x_{0})$ and the estimate of $w$ on $\partial B_{1}$
are subordinate to this estimate, so we conclude the proof. ∎

######  Remark 2.1.

It is straightforward to refine the $|\lambda_{n}|<1$ case and thereby improve
the estimate (1.3) to the following

(2.16) $\displaystyle|Du(0)|\leq
C(n,\|D^{2}\psi\|_{L^{\infty}(B_{1})})(\text{osc}_{B_{1}}u+(\text{osc}_{B_{1}}u)^{2}).$

######  Remark 2.2.

More generally, let $\psi\in C^{1}(B_{1})$ be critical and supercritical, or
$\psi-(n-2)\pi/2=:\phi\geq 0$, and also satisfy the following first order
differential inequality on $B_{1}$:

(2.17) $\displaystyle|D\phi|\leq\eta f(\eta^{-2}\phi),$

where $f(t)\searrow 0$ as $t\searrow 0$, and $\eta=1-|x|^{2}$. Then a $C^{1}$
estimate is valid for $C^{3}(\overline{B_{1}})$ solutions of (1.1):

(2.18) $\displaystyle|Du(0)|\leq C(n,f,\text{osc}_{B_{1}}u).$

To prove the estimate, we insert (2.17) in the determining inequality (2.11)
and use (2.13) and (2.4). We obtain at $x_{0}$, using that $f$ is increasing,

(2.19) $\displaystyle M^{-1}\,\eta|Du|$ $\displaystyle\lesssim
1+\eta^{2}|\lambda_{n}|f\left(\frac{1}{\eta^{2}|\lambda_{n}|}\right)$
$\displaystyle\lesssim 1+\eta|Du|f\left(\frac{C(n)}{\eta|Du|}\right).$

If $\eta|Du|\geq C(n,f,M)=:H$ for large enough $H$ such that
$f(C(n)/H)\ll_{n}M^{-1}$, then $M^{-1}\eta|Du|\lesssim 1$, and the estimate
follows. In the alternative case that $\eta|Du|\leq C(n,f,M)$, the estimate
(2.18) is already done.

######  Remark 2.3.

Let us list some examples of phases which satisfy a first order inequality of
the form (2.17).

1\. For $f(t)^{2}=t^{2}+Ct$, we recover the $C^{2}$ interpolation inequality
(2.12). Note that general $C^{1,\alpha}$ phases fail to satisfy the inequality
(2.17).

2\. Interpolation inequality (2.12) can be generalized to phases $\psi\in
C^{1}(B_{1})$ which are semi-concave, with $D^{2}\psi\leq KI$ for some $K>0$.
In this case, the dependence on $\|D^{2}\psi\|_{L^{\infty}(B_{1})}$ is
replaced with $K$. Indeed, by semi-concavity, there holds for $x_{0}\in
B_{1}(0)$ and $x\in B_{\delta}(x_{0})$:

(2.20) $\displaystyle 0\leq\phi(x_{0})+(x-x_{0})\cdot
D\phi(x_{0})+K|x-x_{0}|^{2}/2.$

The proof in [Hor83, Lemma 7.7.2] can then be repeated verbatim. This
generalizes Theorem 1.1 to semi-concave phases.

3\. The choice $f(t)=2t$ corresponds to $\psi\in C^{1}(B_{1})$ concave.
Choosing $x-x_{0}=-(1-|x_{0}|)D\phi(x_{0})/|D\phi(x_{0})|$ with $K=0$ in
(2.20) gives

(2.21)
$\displaystyle|D\phi(x_{0})|\leq\frac{\phi(x_{0})}{1-|x_{0}|}\leq\frac{2\phi(x_{0})}{\eta}.$

This is the first term in (2.12), so as in (2.14), we obtain $\eta|Du|\leq
C(n)M$. We thus obtain the linear estimate

(2.22) $\displaystyle|Du(0)|\leq C(n)(1+\text{osc}_{B_{1}}u).$

This can be improved to $|Du(0)|\leq C(n)\text{osc}_{B_{1}}u$, as in Remark
2.1. One novelty here is the independence of $\psi$. For example, if

(2.23) $\displaystyle\psi(x)=(n-2)\frac{\pi}{2}+\epsilon(1-|x|^{1+\alpha})$

for some $\epsilon,\alpha\in(0,1)$, then (2.22) is independent of $\epsilon$.
The interior gradient estimate for $C^{1}$ supercritical phases in [Bha21]
would degenerate as $\epsilon\to 0$.

4\. Suppose that $\phi(x)\geq 0$ is a $C^{1}(B_{1})$ viscosity supersolution
of the infinity-Laplace equation/Aronsson’s equation:

(2.24) $\displaystyle D^{2}\phi(D\phi,D\phi)=\phi_{ij}\phi_{i}\phi_{j}\leq 0.$

Then using comparison with cones, there is a pointwise estimate [CEG01, Lemma
2.5] for the gradient:

(2.25) $\displaystyle|D\phi(x)|\leq\frac{\phi(x)}{1-|x|}.$

In fact, this is concavity inequality (2.21), and this corresponds to
$f(t)=2t$ in (2.17). We conclude that a linear gradient estimate (2.22) is
valid.

######  Remark 2.4.

If $u$ is a viscosity solution of (1.1) for Hölder phase

(2.26) $\displaystyle\psi=(n-2)\frac{\pi}{2}+|x|^{\alpha},$

where $0<\alpha<1$, the function

(2.27) $\displaystyle u(x)+C|x|^{2-\alpha}$

is convex, if $C(\alpha)$ is large enough. This follows from the algebraic
relation (2.13), which gives $|\lambda_{min}|<|x|^{-\alpha}$. It follows that
$u(x)$ is locally Lipschitz continuous.

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