# Tagged Questions

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$ ...
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### 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 ...
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### 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 ...
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### 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 $N=(x \frac{d }{dy} -y \frac{d }{dx})$ , ...
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### 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) ...
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### 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.
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### 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 $$\triangle u_1 = C_1(\partial_{ij}u_0),$$ $$\triangle u_0 = C_0u_1,$$ ...
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### 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 ...
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### 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 $u$ ...
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### 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 $R^d$ ($d=2$...
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### 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 ...
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### 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 $T:U \to W$ ...
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### 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 $\partial U$: \...
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### 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 ...