**-1**

votes

**0**answers

27 views

### show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$ [on hold]

For $n\geq 3 $.Let $u\in C^2(R^n), \Delta u\leq 0,u>0$ in $R^n$, show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$.
I was reading the article <Liouville-type theorems and Harnack-type ...

**1**

vote

**1**answer

44 views

### Sobolev regularity for systems of elliptic boundary value problems

My question is about Sobolev estimates near the boundary for elliptic systems (equivalently, elliptic boundary-value problems for vector-valued functions).
Note, results for the scalar case are ...

**1**

vote

**0**answers

44 views

### A priori estimates for elliptic operators

Suppose $L : L^{m,p}(M)\rightarrow L^p(M)$ is some elliptic operator of order $m$, and $(M,g)$ is a compact Riemannian manifold. Then it is known that there exists a constant $C$ such that we have the ...

**0**

votes

**0**answers

92 views

### The eigenfunction of modified $1$-laplace equation?

Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...

**1**

vote

**1**answer

111 views

### Global Poincaré type estimate

For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...

**3**

votes

**3**answers

162 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

66 views

### Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$
...

**3**

votes

**0**answers

70 views

### Global Euclidean Carleman Estimate with a linear phase

I am interested in deriving the following global Carleman estimate which I think should hold :
$ \| e^{\tau \phi} \triangle e^{-\tau \phi} u \|_{L_{\delta+1}^2({\mathbb{R^3})}}> C \tau \| u \|_{L^...

**0**

votes

**0**answers

151 views

### Regularity for a div-curl system

Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...

**2**

votes

**2**answers

308 views

### Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...

**2**

votes

**1**answer

66 views

### What is the function space $H^1_{m, \sigma}$?

I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order".
Theorem 3.1 is the first time in their paper that the function space $...

**1**

vote

**0**answers

45 views

### $L^2$-estimate for an elliptic equation

Given $(M, g)$ a compact Riemannian manifold of dimension $n \geq 3$, we consider the following equation for the function $\phi$:
$$
-\Delta \phi + R \phi + \tau^2 \phi^{N-1} = \frac{A^2}{\phi^{N+1}}
...

**2**

votes

**1**answer

342 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 ...

**0**

votes

**0**answers

12 views

### Setting bound on particular integral when proving properties of Bogovskii operator

I am reading the proof of the properties of Bogovskii operator in the book Introduction to the Mathematical Theory of Compressible Flow.
Let $B^\epsilon(y) = \{ x : |x-y| > \epsilon \}$, $f$ and $...

**0**

votes

**1**answer

83 views

### Bound of solution of pde with a distance function

Hello，i am going to solve the PDE,
$\Delta u = -K in \Omega $and u=0 on the boundary ，where K is some poistive constant.
Then i have read a paper which stated that u>=distance(x，boundary of $\Omega$...

**8**

votes

**1**answer

292 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

**0**answers

66 views

### Elliptic regularity on the hypercube

Assume
$$
Lu=f\quad \text{in } [0,1]^d\\
u=0 \quad\text{ on } \partial[0,1]^d
$$
for some strongly-elliptic operator $L$, and $f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the ...

**1**

vote

**0**answers

52 views

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

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

**4**

votes

**1**answer

440 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 ...

**1**

vote

**0**answers

59 views

### Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...

**3**

votes

**0**answers

59 views

### A priori $C^0$ estimates for a semi-linear vector Poisson equation

Main Question
Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation:
$$
-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0
$$
Here $a_i:...

**0**

votes

**0**answers

20 views

### Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy,
$$
\int_{\Omega}|\nabla u |^2\, -\, \int_{\...

**1**

vote

**0**answers

53 views

### Domain of operator

Let be $\lambda\in C^{*}$. Consider the following operator:
$ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$,
where
$N=(x \frac{d }{dy} -y \frac{d }{dx})$ ,
...

**6**

votes

**1**answer

261 views

### finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed

I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...

**0**

votes

**1**answer

127 views

### a condition for Laplacien

Let $u\in L^{2}(R^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(R^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(R^{2})$?
Thank you in advance.

**0**

votes

**0**answers

65 views

### Solving a system of Laplace equations

Let $u_0$ and $u_1$ to be smooth functions defined on $\Omega\subset\mathbb{R}^n$, consider the following system of equations
$$\triangle u_1 = C_1(\partial_{ij}u_0),$$
$$\triangle u_0 = C_0u_1,$$
...

**5**

votes

**1**answer

194 views

### minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...

**1**

vote

**1**answer

110 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 $u$ ...

**1**

vote

**1**answer

257 views

### Reference to the Existence and Uniqueness of the PDE system

I've the following Problem on systems of Partial Differential Equations. I have "$ N $" Physical variables. and Finally I form the equation on a bounded domain having regular boundary in $R^d$ ($d=2$...

**0**

votes

**0**answers

42 views

### When can an analytical solution for the heat equation be obtained?

I am currently trying to model a system with a time varying heat flux. It seems most researchers are using FEM to obtain the heat distribution (solve the heat equation).
When can the heat equation be ...

**2**

votes

**1**answer

104 views

### Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...

**0**

votes

**0**answers

22 views

### how can we extend this result [duplicate]

Let $T_{a},a\in C$ be a closed operator defined on $D$ subspace of $L^{2}(R)$ onto $L^{2}(R)$ $(T_{a}: D\rightarrow L^{2}(R) )$ with
$D$ contains a Schawrz space $S$
$\Big<\psi,T_{a}\varphi\...

**1**

vote

**0**answers

87 views

### Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.
The most ...

**1**

vote

**0**answers

71 views

### Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....

**7**

votes

**1**answer

117 views

### Properties of connection Laplacian on vector fields

Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...

**0**

votes

**0**answers

61 views

### Order of vanishing of Laplace's equation with potential

Consider the equation $-\Delta u + V u = 0$ with Dirichlet boundary conditions on the bounded domain $\Omega \subseteq \mathbb{R}^n$, where $V$ is a smooth potential. Let $V \leq 0$, and bounded on $\...

**3**

votes

**0**answers

59 views

### On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...

**0**

votes

**0**answers

25 views

### Holder regularity of a singular elliptic problem

In the view of the paper https://pdfs.semanticscholar.org/063b/59c1f070c9b58ab88ed7ef76209187ce7e5c.pdf,
I have a problem about Theorem 1.1 (iv) , I do not understand how $I \leq cd^{1-\alpha}(x_1,...

**3**

votes

**1**answer

138 views

### Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...

**1**

vote

**2**answers

68 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...

**5**

votes

**1**answer

190 views

### How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in H^{1/2}(\partial\Omega)=H^1(\Omega)/H^...

**1**

vote

**0**answers

96 views

### Weyl's law for minimal surfaces

I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...

**5**

votes

**2**answers

143 views

### significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...

**0**

votes

**0**answers

82 views

### Continuity of solutions of nonlinear elliptic PDEs

Consider the nonlinear 2nd order elliptic PDE $$\sum_{i, j} a_{ij}(x, t) \partial_i\partial_j u + \sum_k b_k(x, t) \partial_k u + c u = F(u), \quad x \in \mathbb{R}^n, t \in [0, \infty).$$
Here $a_{ij}...

**1**

vote

**0**answers

28 views

### Maximum principle of the gradient of harmonic extension under weak regularity assumption

I have a question which is very likely to be trivial, but I'm stuck on it! Suppose $f \in W^{1, 2}(B_2(0))$ and $\|{\nabla f\|}_{L^{\infty}(B_2(0)\setminus B_1(0))} < \infty$. Consider then the ...

**0**

votes

**0**answers

62 views

### Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\...

**1**

vote

**1**answer

71 views

### leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...

**2**

votes

**2**answers

150 views

### Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...

**8**

votes

**0**answers

174 views

### Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...

**3**

votes

**0**answers

50 views

### Interior regularity for elliptic operators with non smooth coefficients

I need a pretty standard interior regularity result for a second order elliptic operator of the form
$$
-\nabla^b \cdot (A(x) \nabla^b v)+c v=f,
\qquad
\nabla^b=\nabla+ib(x)
$$
where $A(x)$ is a ...