**1**

vote

**1**answer

234 views

### Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...

**3**

votes

**1**answer

163 views

### On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation
$$\triangle \varphi = 4 \pi \rho \qquad (1)$$
defined on 3-dimensional oriented Riemannian manifold $(M,g)$,
where $g$ is metric and ...

**3**

votes

**1**answer

152 views

### Regularity of Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates
$\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert
_{W^{s,p}}$, for ...

**2**

votes

**2**answers

175 views

### Double-layer potentials on Riemannian manifolds

Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function ...

**0**

votes

**0**answers

12 views

### Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g ...

**3**

votes

**1**answer

310 views

### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...

**2**

votes

**1**answer

143 views

### Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...

**1**

vote

**0**answers

30 views

### Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...

**2**

votes

**0**answers

59 views

### Dirichlet-to-Neumann Map is selfadjoint

Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$.
For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...

**7**

votes

**1**answer

445 views

### $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...

**3**

votes

**0**answers

92 views

### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...

**2**

votes

**0**answers

104 views

### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...

**0**

votes

**0**answers

103 views

### For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?

Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that ...

**1**

vote

**0**answers

22 views

### Elliptic Equation with Wentzell boundary condition

I'm looking for a reference showing how to obtain a priori estimate for solutions to a linear second-order elliptic equation with Wentzell boundary condition in a bounded domain in $H^1$ space.
The ...

**2**

votes

**2**answers

132 views

### References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity.
I studied always, following Evans book "PDE", the case with ...

**0**

votes

**0**answers

93 views

### A priori estimate for diffraction problem for linear elliptic PDEs

I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation.
I looked at ...

**6**

votes

**1**answer

153 views

### Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it.
Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that
$$
...

**0**

votes

**0**answers

35 views

### Regularity result for Neumann problem

I have two questions.
On Elliptic regularity for the Neumann problem, he asked that test function v must be of mean value zero. However, isn't it true that we only need f is of mean value zero and ...

**3**

votes

**2**answers

127 views

### A Global Estimates for Linear Elliptic PDE

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...

**3**

votes

**0**answers

66 views

### Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi ...

**2**

votes

**2**answers

133 views

### Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me.
Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...

**1**

vote

**0**answers

24 views

### Regularity of a flux induced by a potential

Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a ...

**4**

votes

**2**answers

580 views

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**3**

votes

**2**answers

226 views

### What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background:
$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
...

**3**

votes

**1**answer

173 views

### stability of the Monge-Ampère equation

Is there any hope to prove this conjecture (or a similar one)?
Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases}
...

**0**

votes

**1**answer

69 views

### Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says:
Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$.
...

**3**

votes

**1**answer

120 views

### dirichlet problem in the heisenberg group

Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...

**3**

votes

**0**answers

99 views

### Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for ...

**0**

votes

**0**answers

58 views

### How Minimal solution is obtained as limit of approximations

I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this:
I can understand the proof, but I don't know why this solution which is obtained as a ...

**4**

votes

**1**answer

83 views

### Regularity up to the boundary for the Poisson problem

It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...

**1**

vote

**0**answers

96 views

### Limit for eigenvalues of the Dirichlet problem

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, ...

**0**

votes

**0**answers

32 views

### Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference and hint for it.
Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and
let $u$ be a positive distributional supersolution to ...

**5**

votes

**1**answer

66 views

### Modified mean value property

Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant.
I met this modified mean value property of a solution $u$ of $Lu=0$ as
...

**2**

votes

**0**answers

55 views

### Hypoelliptic pseudodifferential operators and Fredholm equations?

I read here
the following:
The parametrix is a useful concept in the study of elliptic
differential operators and, more generally, of hypoelliptic
pseudodifferential operators with variable ...

**1**

vote

**0**answers

92 views

### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} ...

**1**

vote

**1**answer

230 views

### Proof of regularity for bounded elliptic problem

We consider the boundary value problem for potential in the form:
$$-\Delta u(\boldsymbol{x})=0,\quad \boldsymbol{x}\in \mathbb R^3\smallsetminus S,$$
with boundary conditions
$$\nabla ...

**2**

votes

**1**answer

103 views

### Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.
I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...

**2**

votes

**0**answers

181 views

### One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ...

**2**

votes

**1**answer

209 views

### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...

**16**

votes

**4**answers

995 views

### Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**0**

votes

**0**answers

135 views

### Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...

**1**

vote

**1**answer

114 views

### Are solutions of the Beltrami Equations necessarily smooth?

Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The Beltrami Equations are defined as the following system of PDE’s on the domain $ ...

**4**

votes

**0**answers

60 views

### Minimisers and critical points of variational integrals

In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral:
\begin{equation*}
...

**1**

vote

**0**answers

33 views

### Sobolev spaces, Finite Element Error Analysis

Let $\Omega\subset \mathbb{R}^{2}$ be a bounded, convex, polygonal domain and $H=\{(u_{1},u_{2})\in H^{\epsilon-\frac{1}{2}}(\Omega)\times ...

**1**

vote

**0**answers

40 views

### How to define Biharmonic operator for second order sobolev spaces

I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have
$$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : ...

**2**

votes

**2**answers

206 views

### Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian ...

**5**

votes

**1**answer

175 views

### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...

**2**

votes

**1**answer

154 views

### Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...

**1**

vote

**0**answers

72 views

### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & ...

**1**

vote

**0**answers

54 views

### Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...