# 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.

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### Smoothness of solution map for PDE

I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
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### Are there maximum principles related to the third boundary condition?

I am working on this problem \begin{equation} \begin{cases} u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\ u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0, \end{cases} \end{...
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### Ancient Heat equation and Liouville's theorem

I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation: \begin{equation} u_{t} = \...
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### First eigenfunction of the p-Laplacian in an interval $(a,b) \subset \mathbb R$

What is the explicit expression of the first eigenfunction $u$ of the $p$-Laplacian ($p>1$) in a bounded interval $(a,b) \subset \mathbb R$ (up to multiplicative constant)? \begin{equation} \begin{...
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### Parabolic Brezis-Nirenberg problem

Has the parabolic version of the famous Brezis-Nirenberg problem ever been studied?
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### Trace operators on submanifolds

In the following paper, Sobolev Spaces on Riemannian Manifolds with Bounded Geometry: General Coordinates and Traces https://arxiv.org/abs/1301.2539 The authors prove trace theorems for general ...
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### $W^{2,p}$-estimates for Neumann boundary condition to Poisson equation

Consider the following Poisson-Neumann problem in a lipschitz bounded domain $\Omega\subset \mathbb{R}^3$: $-\Delta u=F,\quad \partial_n u\restriction_{\partial\Omega}=0$. Here $F\in L^p(\Omega)$. ...
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Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation $$\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),$$ for some fixed $p\in C^2(\... 0answers 129 views ### Is$(e^{u}+1)\Delta u+u=0$the Euler-Lagrange equation of a functional energy? Does there exist a functional energy$I$such that $$(e^{u}+1)\Delta u+u=0$$ is the Euler-Lagrange equation associated with the energy functional$I$? 0answers 36 views ### A classic uniqueness problem in a constraint minimization problem Consider the following constraint minimization problem $$\inf_{\| u \|_p = 1} \int_{\mathbb{R}^N} |\nabla u|^2 + V(x)u^2 \,dx$$ where$\| \cdot \|_p$is the$L^p$norm,$2 < p < \frac{2N}{N-2}...
Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$. I am interested on the existence of solution for the following problem: given a continuous ...
Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the ...