Questions tagged [elliptic-equations]

Question 1

Let $\Omega \subset \mathbb{R}^n$ a bounded domain, $f:\mathbb{R} \to \mathbb{R}$ of class $C^1$ such that $f(0)=0$. Let $u \in L^{\infty}(\Omega) \cap W_0^{1,p}(\Omega)$ for some $1 \leq p <+\...

Question 2

Lemma 6.33. Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and let $S$ be a bounded subset of the Banach space $$C_*^{k, \alpha} = \left\{ u \in C^{k, \alpha}(\Omega) \mid |u|_{k, \alpha; \Omega}...

Question 3

NOTICE TO ALL: I do NOT want a solution to the below exercise. Please do not post one. I am working on that myself and don't want it spoiled. I am working on the exercises from chapter 6 of Evans ...

Question 4

I am reading Evans "Partial Differential Equations" (2nd Ed) and going through the proof of a specific case of the general Harnack inequality. In the following, $U\subset\Bbb R^n$ is open ...

Question 5

I am currently reading chapter 6 of Evans "Partial Differential Equations" and have hit another small roadblock. I am working through the theorem in 6.5.1 , which I reproduce below: THEOREM ...

Question 6

I'm reading chapter 6 of Evans "Partial Differential Equations" (2nd Ed) and I am reading section 6.5.2, "Eigenvalues of nonsymmetric elliptic operators". Here we are studying ...

Question 7

I wonder if the spectrum of linear 2nd elliptic operator $$Lu=-\Delta u + \eta \cdot \nabla u + hu$$ on bounded domain is discrete. Here $h$ is a smooth function without sign condition and $\eta$ is ...

Question 8

In the linked question, the poster claims that distributional solutions of the Helmholtz equation, $$-\Delta u+u=f ~~~ \text{in}~\Bbb R^n$$ Satisfy $u\in W^{2,p}(\Bbb R^n)$ whenever $f\in L^p(\Bbb R^n)...

Question 9

I am considering the following problem from an introductory course on elliptic PDE theory. 'Suppose that $\Omega$ is a domain in $\mathbb R^n$ (that is to say, a non-empty connected open subset of $\...

Question 10

I'm reading a paper where they do following claim: Let $u \geq0$ a function in $H^1(\mathbb{R}^2)$ and $f \in C^1(\mathbb{R})$ such that $f(s)=0$ for $s \leq0$ and $F(s)=\int_{0}^{s} f(t)dt$. In this ...

Question 11

Let the Dirichlet problem $$\begin{cases} -\operatorname{div}(A(x)Du)=f,\,\Omega,\\ u=0,\,\Omega^c \end{cases},$$ where the operator $A(x):\mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$ verifies ...

Question 12

Let $M$ be a compact Riemannian manifold with connected smooth boundary $\Sigma$. Let $\Sigma_D$ and $\Sigma_N$ be two disjoint smooth domains of $\Sigma$ with $\Sigma = \overline{\Sigma_D\cup\Sigma_N}...

Question 13

Let $T>0$, $v\in C^{2,\alpha}(\mathbb{R}/T\mathbb{Z})$ whose norm is small enough, where $\mathbb{R}/T\mathbb{Z}$ denotes the circle of perimeter $T$. Define the perturbed cylinder $C_{1+v}^T$ by $$...

Question 14

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary (for Sobolev embedding and elliptic regularity). The $H^2$ norm is: $$ \|u\|_{H^2(\Omega)}^2 = \|u\|_{L^2(\Omega)}^2 + \|\...

Question 15

Let $\Omega\subset \mathbb{R}^N$ be a bounded smooth domain. I am interesting in the existence of $W^{1,p}_{0}$-weak solution of the problem $$\begin{cases}\Delta u+\lambda u = f&~\text{in }\Omega\...

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Questions tagged [elliptic-equations]

Composition in Sobolev Spaces [duplicate]

Lemma 6.33 in Gilbarg & Trudinger's book

Admissible test functions for weak formulation of Neumann problem

Detail in proof of general Harnack inequality (Evans PDE)

Small detail in proof of "Eigenvalues of symmetric elliptic operators" (Evans PDE 2nd Ed)

What is a "symmetric" elliptic operator?

The discreteness of spectrum of linear 2nd elliptic operator $Lu=-\Delta u + \eta \cdot \nabla u + hu$.

Reference for regularity result claim for Helmholtz equation when $n>2$

Unique continuation for elliptic partial differential equation with C^1 coefficients: an elementary approach?

Question about Schwarz symmetrization

Showing continuity of operator $a:W_w^{1,2}\times W_w^{1,2}\to \mathbb{R}$

$H^{\frac{3}{2}}$-regularity of a specific elliptic mixed boundary valued problem.

Uniqueness of radial solution on the perturbed cylinder

Equivalent norm on $H^2(\Omega)$

Existence of $W^{1,p}$-weak solution to $\Delta u + \lambda u = f$ when $\lambda$ is an eigenvalue

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