# Questions tagged [elliptic-pde]

Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

1,013
questions

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### An estimate of the gradient of heat kernel

We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I have already proved that
$$
\...

1
vote

0
answers

27
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### Regularity of a weak solution to an elliptic PDE with mixed boundary condition

I have a question on the regularity of a weak solution to an elliptic PDE.
Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ and $r>0$ satisfy $B(x,r)\...

2
votes

0
answers

43
views

### Maximizing the first Neumann eigenvalue on disks

Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that
$$\mu_1(g) \operatorname{...

2
votes

1
answer

70
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### Smooth dependence of parameter of PDE - viscosity solutions

There is a prevalent method called the "Nonlinear adjoint method" in the study of viscosity solution and Hamilton--Jacobi equation, especially equations of the form
$$
u^\varepsilon + H(x,Du^...

0
votes

0
answers

112
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### Generalized harmonic map

Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as
$$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$
When $c=0$, ...

2
votes

0
answers

62
views

### Strong maximum principle for weak solutions still holds?

By De Giorge, Nash and Moser solutions of
\begin{equation}
\operatorname{div} (A(x) Du) = 0
\end{equation}
where $Du$ denotes the gradient of $u$ and $A$ is a $\lambda,\Lambda$ elliptic matrix. ...

3
votes

0
answers

34
views

### Intersection of $n$-dimensional minimal surfaces with two-dimensional planes

Let $M^n \subset \mathbf{R}^{n+k}$ be a smoothly embedded minimal surface. When the dimension is $n = 2$ and the codimension is $k = 1$ the intersection of $M$ with planes is well understood. If $M$ ...

2
votes

0
answers

73
views

### Test function in Simon Brendle's paper about Yamabe flow

I'm recently reading Simon Brendle's Convergence of the Yamabe flow in dimension 6 and higher.In this paper,he constructs a family of test functions with Yamabe energy less than Yamabe constant of ...

3
votes

1
answer

123
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### 'Degenerate' tangent point of a minimal graph

Let $u: D_1 \to \mathbf{R}$ be a smooth function defined on the unit disk $D_1 \subset \mathbf{R}^2$ which describes the minimal graph $G$. Suppose that at the origin $G$ is tangent to the horizontal ...

4
votes

1
answer

155
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### Approximate isometric embeddings of surfaces

The fundamental theorem of surfaces states that if symmetric matrices $g_{ij}$, $l_{ij}\colon U\subset R^2\to R$, where $U$ is open and $g_{ij}$ is positive definite satisfy the Gauss and Codazzi ...

8
votes

3
answers

469
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### Regularity of Newtonian potential along smooth boundary

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...

2
votes

0
answers

47
views

### Well-posedness or existence for a Poisson problem in Orlicz spaces

I know that the problem
\begin{equation}
\Delta_p u = f
\end{equation}
make sense if $f \in L^q$ with $n/p<q<n$ and that is there a existence theory for
$$
u_t -\Delta_p u = f
$$
For a given ...

1
vote

1
answer

50
views

### Viscosity characterization of convex functions

Let $\Omega\subseteq\mathbb{R}^n$ open and convex. It is elementary that if $u\in C^2(\Omega)$ then
$$u \text{ is convex}\iff D^2u\geq0 \ \text{ in } \Omega$$
I was looking for a similar ...

2
votes

0
answers

61
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### Spectrum of 'complexified' Laplace operator

Let $(M^n,g)$ be a closed Riemannian manifold. Let $\Delta$ be the Laplace–Beltrami operator acting on scalar functions defined on $M$, and let
$\lambda_1 < \lambda_2 \leq \cdots$ be its spectrum.
...

1
vote

0
answers

71
views

### Nonlocal elliptic problem - what is its associated energy?

It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...

3
votes

1
answer

147
views

### On a compact operator in the plane

Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$
and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...

8
votes

2
answers

465
views

### Compactly-supported harmonic tensors

Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$...

0
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0
answers

37
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### Gehring lemma for fractional maximal functions

Given a function $f\in L^p(\mathbb{R}^n)$, the Gehring lemma states that if there exists $p>1$, a constant $C_0>1$ and a cube $Q \subset \mathbb{R}^n$ such that for almost every $x\in Q$, it ...

1
vote

0
answers

83
views

### For the solvability of the poisson equation $\Delta u = f$ on manifold with boundary

For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...

2
votes

0
answers

90
views

### A question about Gauss-Green formula - a weaker assumption

The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place
$$\...

4
votes

0
answers

103
views

### Eigenvalues of Schrödinger operator with Robin condition on the boundary

Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...

2
votes

0
answers

63
views

### Approximating solutions to Monge-Ampere from optimal transport plans

I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...

4
votes

1
answer

102
views

### approximation of a Feller semi-group with the infinitesimal generator

Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...

2
votes

0
answers

35
views

### Elliptic regularity and compact embedding in a weighted Sobolev space

Let $B \subset \mathbb{R}^3$ be the open unit ball centered at $0$ and consider the weight function $w(x)=|x|^2$. Suppose $u \in C_0^\infty(B)$ and consider weighted Sobolev $H^2_w(B)$ norm
$$\|u\|_{H^...

2
votes

0
answers

99
views

### Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply.
When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...

2
votes

0
answers

31
views

### A mapping property for fractional Laplace--Beltrami operator

Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...

2
votes

1
answer

92
views

### Global Hölder regularity

I am reading the book "Regularity theory for elliptic PDE" by Xavier Fernández-Real
and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $...

1
vote

1
answer

102
views

### comparison principle for the minimal surface equation

Consider the (inhomogeneous) minimal surface equation for functions $u,f:D\to \mathbb{R}$ for some smooth domain $D\subset \mathbb{R}^n$
$$Lu:=\operatorname{div} \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=...

2
votes

0
answers

99
views

### Linear elliptic problems: Are gradient estimates preserved after perturbation?

(This question is a duplicate from here)
We start with the linear elliptic PDE
$$
-\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\
u=0 \quad\text{on}\ \partial\Omega
$$
We assume that $\...

1
vote

0
answers

72
views

### positivity of solution for Baouendi-Grushin operator

Let $(x,y)\in\mathbb{R}^N = \mathbb{R}^m \times \mathbb{R}^n$ with $m \geq 1$, $n\geq 0$, $\alpha \geq 0$ and $ Q=m+ n(\alpha +1)$.
Consider the following eqution
\begin{equation}\label{critical}
-(\...

4
votes

0
answers

89
views

### Continuity of solutions of Elliptic PDE with respect to parameters

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...

1
vote

1
answer

99
views

### Dense properties of weighted Sobolev space define on $\mathbb{R}^n$

Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....

2
votes

1
answer

100
views

### Existence of eigen basis for elliptic operator on compact manifold

Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting ...

2
votes

0
answers

79
views

### Critical points of a strictly subharmonic function

Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant:
\begin{equation}
\Delta u = A > 0....

10
votes

1
answer

553
views

### Hodge decomposition in elliptic complexes

EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a ...

1
vote

0
answers

87
views

### About Agmon-Douglis-Nirenberg complementing boundary condition

Let's consider the following Poisson equation with Neumann boundary condition
\begin{align*}
-\Delta u &= f \quad \text{in } \Omega \\
\partial_n u&=0 \quad \text{on } \partial \Omega,
\end{...

3
votes

0
answers

85
views

### Comparison principle for Elliptic PDE with exponential nonlinearity

Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...

5
votes

3
answers

313
views

### Poisson equation on manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...

1
vote

0
answers

54
views

### Poisson equations for tensors on compact Riemannian manifold

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation
$$\Delta f=S$$
where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...

7
votes

1
answer

204
views

### Existence and estimates of Green's function on Riemannian manifold

In Yau and Schoen's differential geometry,in Ch5 before Thm 3.5,the author says
When $R$(scalar curvature of a manifold M)$>0$,there exists a unique Green's function $G$ to the operator $L=-\Delta+...

0
votes

0
answers

51
views

### Estimate value of harmonic function in the annulus

Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $Lu = 0$ in $D$ where $L$ is a uniform elliptic operator with constant coefficients
$$
Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...

3
votes

0
answers

92
views

### What notion of weak solution is suitable for systems of $\infty$-elliptic PDE?

Let $Pu = f$ be an elliptic PDE in divergence form. Then $P$ is viewed as a generalization of the Laplacian, and we can define its weak solutions analogously to how we define a weakly harmonic ...

2
votes

0
answers

92
views

### A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define
$$
H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N.
$$
By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...

1
vote

1
answer

155
views

### Integral identity for critical points of the Ginzburg-Landau functional

I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional
$E_\epsilon(v) = \frac{1}{2}...

1
vote

0
answers

56
views

### Limiting behavior of Kazdan-Warner equations

It's a well known result by Kazdan and Warner that on a closed Riemannian manifold the pde:
\begin{align*}
\Delta f+ge^f=c
\end{align*}
has a unique solution for $g\geq 0,$ and $c$ a positive constant....

1
vote

1
answer

139
views

### Higher integrability for Sobolev functions - part 2

This is a follow-up to the question asked in Higher integrability for Sobolev functions
Updated question: Given the very helpful counterexamples and the ideas, I have the following question: Suppose ...

5
votes

3
answers

547
views

### Higher integrability for Sobolev functions

Let $u \in W^{1,2}(\mathbb{R}^2)$ be a given function satisfying $$\frac{1}{|B_r|}\int_{B_r(x)} |\nabla u|^2 dy \leq \frac{1}{r^{\delta}}$$ for all $r \leq 1$, $x \in \mathbb{R}^2$ and some fixed $\...

7
votes

0
answers

64
views

### Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...

0
votes

0
answers

23
views

### Transformation of domain under uniformly elliptic matrix

Let $A(x)$ be a smooth, uniformly elliptic, symmetric matrix, i.e., $\lambda |\xi|^2 \leq \langle A(x)\xi,\xi\rangle \leq \Lambda |\xi|^2$ for some two fixed constants $\lambda, \Lambda \in (0,\infty)$...

1
vote

0
answers

45
views

### Dirichlet-to-Neumann estimate for minimal graphs

Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation
\begin{equation}
(1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0
\end{...