On an effective equation of the reduced Hartree-Fock theory
-
Ilias Chenn
,
Wei Wang
Abstract
We show that there is a one-to-one correspondence between solutions to the Poisson-landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.
1 Introduction
1.1 Reduced Hartree-Fock equation
Despite the success of the density functional theory (DFT), its computational difficulties remain a major bottleneck. Filoche and Mayboroda initiated a series of recent works on the landscape function [18], which led to a further simplification of the DFT by introducing the Poisson-landscape (PL) equation [19,37]. The landscape theory and numerical simulations [2–4,19,37,43] suggest that solving the PL equation can be an efficient and accurate replacement of the original DFT. This success undoubtedly demands a rigorous mathematical justification and a theoretical foundation.
DFT originated as a systematic way to study the large many-body quantum system by using a self-consistent 1-body approximation. Parallel to its development, a number of effective theories existed along with DFT; examples include the Hartree-Fock theory, the Bardeen-Cooper-Schrieffer (BCS) theory, and the Thomas-Fermi theory of electrons. While DFT enjoyed a similar energy functional as the more complex Hartree-Fock theory and the BCS theory, inheriting a form of accuracy, it also gravitated toward the Thomas-Fermi theory to study the simpler electron density instead of density matrices. Owing to these characteristics, the Kohn-Sham (KS) energy and the equation of DFT were developed [21,22]. These equations and their related theory have become a mainstay of modern condensed matter physics. Some notable areas of application include semi-conductor design, deformation theory in solid mechanics, and quantum chemistry. In the mean time, a plethora of mathematical studies also ensued, for example, see [17,23,26,27,30–33,35,42].
The KS equation is a set of functional equations for the electron density
Consider a semi-conductor at a positive temperature,
where
and
where
We remark that while equation (1) is an equation for microscopic electronic structures of matter, dopant potentials and band-offsets often vary on another larger mesoscopic scale. A precise formulation of the problem would require a homogenized version of (1) where mesoscopic parameters such as the dielectric operator emerge. However, we will make the possibly unphysical assumption that (1) is already homogenized and the dielectric constant is 1 purely for mathematical simplicity (Remark 1.6).
Moreover, we further restrict ourselves to the semi-classical regime and modify (1) as follows:
where
where
1.2 Landscape theory and the PL equation
In one view, the landscape theory presents a partial diagonalization of the Schrödinger Hamiltonian
and the landscape potential
Conjugating
We remark here that
where
This equation was proposed as a computational simplification to the REHF and studied in the physical work [19,34,37]. Together with (3), they bear the name PL equation.
Numerical solution to the REHF equation requires an extensive computation of a large number of eigenvalues and eigenfunctions of the Hamiltonian
where
numerically. This approximation enjoys a more accurate prediction than the usual Weyl’s law on average. We note that the left-hand side of (15) is
where
Up to now, many of the stated advantages of the landscape theory have mostly been proven useful for numerical purposes. The current article is the first rigorous mathematical treatment of the PL model. The goal of the current work is to introduce a rigorous treatment of the PL equation as an effective equation of the REHF equation in the semi-classical limit. Other related rigorous mathematical treatments of the landscape theory for different models can be found in [3,5,16,39,43].
1.3 Results
We limit ourselves to the periodic setting in which physical quantities are periodic on
Moreover, let
We let
Definition 1.1
Let
where
Our assumption on the external potential
where
Throughout the article, we will write
Our first result shows that the density on the right-hand side of (6) can be approximated by the right-hand side of (11). This result will be proved in Section 3.
Theorem 1.1
Let V be a Landscape admissible potential given in the form (19). In addition, assume that
and two effective potentials
Moreover, the density has the asymptotic expansion
where
Theorem 1.1 provides the foundation for a rigorous justification of the PL equation. In addition, (22) suggests that a simpler effective equation is also possible. More precisely, let
where
One advantage of
where
Hence, all the material and doping properties are stored in
Finally, to state our main result relating the REHF, PL, LSC equations, and the associated electric fields, we specify additional assumptions.
Assumption 1
(Semi-classical regime). The semi-classical parameter
for some large constant
Assumption 2
(Low temperature). There is some
Remark 1.2
The positive temperature assumption
Assumption 3
(Conservation of charge). The doping potential
is a fixed constant.
Theorem 1.3
(Main result) Let assumptions in
Theorem 1.1
and
Assumptions 1–3
hold. Assume that
Then there exists
Theorem 1.3 has an immediate corollary in terms of the density
Corollary 1.4
Retain the assumptions in
Theorem 1.3. Assume that
Then there exists
Remark 1.5
Corollary 1.4 answers the challenge posed in the introduction. It justifies [19, 37] on a mathematically rigorous level in the semi-classical regime at low temperature (or large
Remark 1.6
We noted in the paragraph before equation (6) that the dielectric constant is taken to be 1. However, as one will see from the proof of our main result Theorem 1.3, so long as the dielectric constant is strictly positive, the same conclusion can be derived, albeit with more cumbersome proofs.
Remark 1.7
Note that in both Theorem 1.3 and Corollary 1.4, a solution to (29) is a pair: either
(Dilation symmetry)
(39)for
(Complex conjugation) If
(40)is a symmetry of (29) where
One also note that the PL equation with (27) does not have the dilation symmetry, contrasting the case of (26) and (28). Whether this difference makes numerical approximations using (27) less desirable is out of the scope of this article, since (27) respects the dilation symmetry in leading order
Theorem 1.3 could help us to prove existence of solutions for the three classes of equations REHF, PL, and LSC simultaneously. However, we were unable to prove the smallness assumption (33) in general. However, we believe this condition should hold in many cases if
Theorem 1.8
(LSC existence) If
Proof
This is a direct corollary of Theorem B.1.□
1.4 Outline of the proof
The proof consists of mainly two parts. The first part is an leading order expansion of electron density,
The second part is to use Theorem 1.1 to prove Theorem 1.3, relating the REHF, PL, LSC equations, and the associated electric fields. To do that, we digress briefly in Section 4 to establish a relationship between the parameters
obtained by integrating (29) over
2 Landscape function in the semi-classical regime
In this section, we will obtain several estimates for the landscape function
on
Theorem 2.1
Let
where C depends on d and p only.
We start by estimating
Proposition 2.2
Retain the definitions in
Theorem 2.1. If u solves (42) with periodic boundary condition on
Proof of Proposition 2.2
The first inequality (45) follows from the fact that the Hamiltonian
Notice that
It is easy to check that
First, we prove (46) for
We rewrite
where
For any
Thus, on
for some constant
It follows that
By combining (53) and (54), we see that
Next, we decompose
where
This provides an energy lower bound. On the other hand,
It follows that
This is an energy upper bound. We now estimate the term
Together with (51), we note that
By using (50) and applying Young’s inequality to (52), we see that
By applying (60) and (62) to (61), we see that
Thus, on
For any
We remark that the domain of integration is in fact
Since
We perform Hölder’s inequality (in the
to obtain
By inserting this into equation (67), we obtain
Combining the estimates on
Together with (57) and (59), and using
Subtracting the first two terms of (72) from both (72) and (70), we see that
Therefore,
By combining with (55) and (56), we see that
Finally, we estimate the
Therefore,
Proposition 2.3
Retain the definitions in Theorem 2.1. If u solves (42) with periodic boundary condition, then
where C depends on d and L only.
Proof of Proposition 2.3
Define
Consequently, we estimate the sup-norm of
Let
It follows that
We claim that
for some constants
since
Now we prove the claims (81) and (82). Equation (81) is standard. For the sake of completeness, we carry out the corresponding estimates. Let
It follows from Young’s inequality that
for some constant
Since
for some universal constant
Finally, using
we see that
By equations (79) and (81), and the fact
for some constant
Proof of Theorem 2.1
Since
We now interpolate between equations (46) and (75). Recall
This proves (43). By differentiating
Since
equations (75), (76), and (88) show that
Similarly, equations (46), (75), (76), and (88) show that
By combining (90) and (93) and using (89), one can compute that
This proves (44).□
3 Leading-order expansion of electron density
We first state a more general theorem from which Theorem 1.1 follows. Then, we prove Theorem 1.1 while delaying the proof of the more general theorem until the end of the section. Let
We have the following general result.
Theorem 3.1
Let
Suppose that f is analytic on
(96)uniformly in
The parameters
- (97)
where
for some p-dependent constant
Proof of Theorem 1.1
Let
We note that the Fermi-Dirac function has poles on the imaginary axis in
where
Thus,
To prove (23), we apply Theorem 3.1 with the potential
By the second item of Theorem 1.1 and the Sobolev inequality,
Item 5 of Theorem 3.1 can also be satisfied since
It follows by Theorem 3.1 that the
The remainder of this section is devoted to the proof of Theorem 3.1.
Proof of Theorem 3.1
First, we remark that the potential functions
Let
Let us denote
Consequently,
Since
where the contour
For simplicity, we will denote
for the rest of the article. Let
Then,
where
To extract leading orders and for
It follows from (111) that
Translation invariance of
By using Cauchy’s formula and Taylor’s theorem, the first term
Recalling that
The following Lemma is the main work horse in this estimate, whose proof is delayed until the conclusion of the proof of Theorem 3.1. The Lemma involves the Schatten
We identify the complex plane
Lemma 3.2
Let
for some
Assuming Lemma 3.2, we complete the proof of Theorem 3.1. Let
provided
where
Proof of Lemma 3.2
We will prove (123) only. The proof for (122) is similar. For
We expand the resolvent using the resolvent identity
We will consider each of the
Since commutator of
where h.o. will be computed after this paragraph. Summing over
Now we compute the higher order terms coming from
We denote
(134)Note that
(135)We denote
(136)(137)These terms came from the commutator
(138)when we commute
We write
If we write
where
It follows that
where we note that the leading term
First, we estimate the Schatten
Lemma 3.3
Let
where
Proof
We compute
Kato-Seiler-Simon inequality shows that
Recalling definitions (136) and (137), we see that
By definition (109) of
By applying Lemma 3.3 to equation (143), we see that
This concludes the estimate for (143).
We can estimate (144) in the spirit of the estimate (143) by using an analog of (152). We obtain
Finally, we estimate (145). The term
where
Combining (152) (for (143)), (154) (for (144)), (158) (for (145)), and using the binomial theorem (or a modification thereof), the fact
4 Consequence of the integrability condition
Before we move on to the proof of the main result Theorem 1.3, we dedicate this short section to elucidate the implied relationships between different parameters
Let
The goal of this section is to prove bounds on
Lemma 4.1
Let V be a bounded potential as in (19) and
Proof
We consider the special case where
where
Let
Since
Thus, we see that
where we recall that
for some constant
Corollary 4.2
Let
Assumptions 1–3
hold. Assume also that (161) holds and
for some constant
Proof
By Theorem 1.1, it suffices for us to assume that (163) hold. Lemma 4.1 shows that
where we recall that
5 Proof of the main result: Theorem 1.3
Proof of Theorem 1.3
Recall that we can write the REHF, PL, and LSC equations (29) in the form
For a fixed choice of
Rearranging, we obtain
where
Theorem 1.1, and the scaling in Assumption 1 and 2 show that
for some constant
We see that (179) can be written as follows:
where
The rest of the analysis rests upon the following abstract lemma and subsequent theorems.
Lemma 5.1
(Main lemma) Let
and
for some constant
then there exists a unique solution
to the equation
Moreover,
Proof
This is just the implicit function theorem with explicit estimates written out. See, e.g., Chapter XIV of [24].□
Let
Theorem 5.2
Let the assumptions of
Theorem 1.3
hold. Then L is bounded below on
for some constants C and any
Let
Theorem 5.3
Let the assumptions of Theorem 1.3 hold. The nonlinear operator N has the following estimate
for
Theorems 5.2 and 5.3 are proved in Sections 6 and 7, respectively. Section 4 provides some preliminary estimates on parameters
Now we apply Lemma 5.1 to (184). We take
for some constant
where
by Assumption 1. This proves (189) of Lemma 5.1. Consequently, Theorem 1.3 is proved by Lemma 5.1. We remark that the reality of
6 Linear analysis
In this section, we prove Theorem 5.2 in three parts: in each of the following subsections, we prove a version of Theorem 5.2 for the case of REHF, PL, and LSC in Theorems 6.1, 6.8, and 6.9, respectively.
6.1 Proof of Theorem 5.2: REHF case
Let
be the Gâteaux derivative of
Theorem 6.1
Let the assumptions of
Theorem 1.3
hold. Then
for some constant C.
We begin by recording a few auxiliary lemmas first.
Lemma 6.2
Let the assumptions of
Theorem 1.3
hold. For any
where
Moreover,
for some constant C.
Proof of Lemma 6.2
We will only prove (203). The rest of the properties are proved in [13]. We remark that
where
where
Let
Since
where we note that the additional factor of
Let
where the square root is taken via the Borel functional calculus under periodic boundary condition on
Lemma 6.3
Assume that
In view of Lemma 6.3, define
where
We remark that when
for
Lemma 6.4
Let the assumptions of
Theorem 1.3
hold. Let
for some constant C. Consequently, picking
Proof
We start with equation (201). Let
where
and define
We first consider the leading-order term on right-hand side of (219).
On
By inserting (223) into (222), we see that
For any real numbers
It follows by Taylor’s theorem that
By Fourier transforming back to the position basis, we have that
where
and
where
We now estimate the error terms in (220) and (221). We only consider the term
and the other terms in (220) and (221) are similar. By Lemma A.1, for any
Since we chose
where the extra factor
Since
Corollary 6.5
Let the assumptions of
Theorem 1.3
hold. If
for some constant C and
Proof
We will use the notations
Since
Let us denote
In this notation,
Define
Since
Corollary 6.6
Let the assumptions of
Theorem 1.3
hold. If
for some constant C and
Now we are ready to prove Theorem 6.1.
Proof of Theorem 6.1
Let
Lemma 6.4 and Corollary 6.5 show that
for
It follows that
By Corollary 4.2, Lemma 6.2, and equation (242), we see that
where
To complete the proof of Theorem 6.1, we need the following Lemma, whose proof is delayed until after the current proof.
Lemma 6.7
By combining equation (246) and Lemma 6.7, we see that
By Assumption 1, the proof of Theorem 6.1 is complete, modulo the proof of Lemma 6.7, which we shall provide in the immediate paragraph that follows.□
Proof of Lemma 6.7
Let
Otherwise, we assume that
By elementary calculus, we have the following two estimates
Equation (247) becomes
We can compute the bracketted term explicitly as a function of
This proves (247).□
6.2 Proof of Theorem 5.2: PL case
Let
where
Let
Theorem 6.8
Let the assumptions of
Theorem 1.3
hold. The linear operator L given in (185) with
where
for some constant C, where
Proof
We differentiate
This proves (260).
Let
where
Recalling that
and
Thus, we see that
It follows by the Sobolev inequality, Theorem 2.1, Corollary 4.2, and definition (193) that
Next, we provide a lower bound for
We denote
where
It follows from (280), Theorem 2.1, and Sobolev’s inequality (which depends on
for suitable and possibly different constants
6.3 Proof of Theorem 5.2: LSC case
Similar to the PL case, let
where
where
Theorem 6.9
Let the assumptions of
Theorem 1.3
hold. The linear operator L given in (185) with
where
for some constant C, where
Proof
We differentiate
where
By Corollary 4.2, Assumptions 1 and 2, we see that
7 Nonlinear analysis
In this section, we prove Theorem 5.3 in three parts: in each of the following subsections, we prove a version of Theorem 5.3 for the case of REHF, PL, and LSC in Theorems 7.1, 7.3, and 7.2, respectively.
7.1 Proof of Theorem 5.3: REHF case
Theorem 7.1
Let the assumptions of Theorem 1.3 hold. The nonlinear operator N (defined in (184)) of the REHF equation has the following estimate:
for
Proof of Theorem 7.1
By (184), we see that
where
(recall from Theorem 1.1 and (20) our choice of
By using the Cauchy-integral, the resolvent identity, and (201), we arrive at an explicit formula for
where
By applying the resolvent identity to (296) iteratively with
whenever the series converges, which we will demonstrate. Let
denote the
To do so, we use the following expansion of
By using this pattern, we see that
where
It follows that
by the scaling in Assumptions 1–2 and Corollary (4.2). Denote
By combining with (297) and the assumption that
7.2 Proof of Theorem 5.3: LSC case
We will prove the simpler LSC case first before embarking on the tedious yet similar proof of the PL case.
Theorem 7.2
Let the assumptions of
Theorem 1.3
hold. Let
where
Proof
Let
By Assumptions 1 and 2, we may replace
Since
7.3 Proof of Theorem 5.3: PL case
Theorem 7.3
Let the assumptions of
Theorem 1.3
hold. Assume that
where
Proof
Let
Let
Again we replace
Direct computation shows that
and
Hence,
As mentioned earlier, let
for some constant
Assumptions 1 and 2 allow the
Acknowledgments
The authors are grateful for stimulating discussions with D. N. Arnold, J.-P. Banon, M. Filoche, D. Jerison, and A. Julia. The first author thanks I. M. Sigal for many helpful insights and guidance. The authors also want to thank the anonymous referees for important suggestions and remarks.
-
Funding information: Chenn was supported in part by a Simons Foundation Grant 601948 DJ and a PDF fellowship from NSERC/Cette recherche a été financée par le CRSNG. Mayboroda was supported by NSF DMS 1839077 and the Simons Collaborations in MPS 563916, SM. Wang was supported by Simons Foundation grant 601937, DNA. Zhang was supported in part by the NSF grants DMS1344235, DMS-1839077, and Simons Foundation grant 563916, SM.
-
Conflict of interest: The authors declare that they have no conflict of interest.
Appendix A Trace per volume and associated Schatten norm estimates
Let
An operator on
Let
where
where
The
fails to hold for
Given an operator
for any
then,
whenever
Lemma A.1
Let
where
Proof
We prove the case for
where
where
B Existence of solution to the PL equation
We show that the LSC equation (see (28) and (29)) has a solution by minimizing its associated energy functional. To this end, let
whenever the integral is convergent. Otherwise we set
The associated space on which we perform our minimization is
where, using the notation
We note that
and define projections
For simplicity, we will denote
interchangeably.
Theorem B.1
(LSC existence and uniqueness) Assume that
Proof of Theorem B.1
We equip
Note that
Step 1: Euler-Lagrange equation. Let
From this, the Euler-Lagrange equation subject to
for a suitable
By integrating equation (A25) with respect to
Finally, set
Step 2: Coercivity. Note that without the interaction term (Coulomb term), an unconstrained minimizer to
is
This shows that
for
Step 3: Convergent subsequence. Since
In particular,
for
So we will denote by
Taking
By the same reasoning with
Next, to see that
Since
It follows that
It remains to show that
Let
By Hardy-Littlewood-Sobolev or Sobolev-Poincare and since
Since
It follows by the Riesz representation theorem that
Finally, Hardy-Littlewood-Sobolev or Sobolev-Poincare shows any
Weak convergence of
In summary,
Step 4: lower semi-continuity. By using (A23), we write
From this expression, it is evident that
Step 5: conclusion. By Steps 2 and 3, we see that there is a minimizing sequence
Since
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Articles in the same Issue
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- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
