Questions tagged [regularity-theory-of-pdes]

The concept of regularity concerns the smoothness of weak solutions to partial differential equations.

439 questions
1answer
8k views

What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
2answers
6k views

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check ...
2answers
351 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
1answer
137 views

Sobolev embedding for the $L^q$ norm.

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
1answer
134 views

Is a Sobolev map with smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $d>2$. Let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$ where $\Omega$ is an open subset of $\mathbb{R}^d$. Let $2 \le k \le d-1$ be fixed. Suppose that $\det df>0$...
1answer
275 views

The Schwartz function and the sobolev space $W^{2,p}$

How do you prove the Schwartz functions in $\mathbb{R}^n$ are dense in the space $W^{2,p}(\mathbb{R}^n)?$ Terrence tao has a version of the proof of The space $C_c^{\infty}(\mathbb{R}^d)$ of test ...
0answers
507 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
0answers
123 views

Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
2answers
142 views

$\Delta u=3u$ then $u\equiv0$

I have the following question in which it is easy to use Fourier transform to get the answer if the function is nice enough, for example $u\in C_{0}^{\infty}(\mathbb{R}^{n})$, however here $u$ is only ...
2answers
239 views

Need good reference or a proof on regularity of solution to Neumann problem

Let $\Omega\subset \mathbb{R}^d$ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
1answer
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Is being in the Sobolev space of power $\frac{d}{2}$ necessary for having well defined point evaluations?

From the Sobolev embedding theorem we know that for $\alpha = \frac{d}{2}$, $W^{\alpha, 2}(\mathbb{R}^d)$ is continuously embedded in $C^0(\mathbb{R}^d)$. Especially the point evaluations are in the ...
0answers
29 views

When Sobolev maps are localizable?

Let $M,N$ be smooth $d$-dimensional Riemannian manifolds. A Sobolev map $f \in W^{1,p}(M,N)$ is called localizable if for every $x_0 \in M$ there exists a neighbourhood $U$ of $x_0$ in $M$ and a ...
0answers
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Mean curvature flow - initial condition - mean-convex

The mean curvature flow of a surface given by a graph $X : B \subset \Bbb{R}^n \to [0,\infty)$ is given by $$X_t (x,t) = H(x,t) \vec n(x,t)$$ where $H$ is the mean curvature and $\vec n$ is the ...
3answers
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1answer
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New function defined by the trace of a $H^1$ function

Let $\Omega:=B(0,1)\subset \mathbb R^N$ the unite ball and $N\geq 2$. Given $u\in H^1(\Omega)$. Then the trace $T[u]$ is well defined over $\partial \Omega$. (by $H^1$ I mean $W^{1,2}$ space) Now let'...
0answers
115 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 $\Delta$...
0answers
150 views

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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$L^p$ regularity for weak solutions of non homogeneous heat equation

Let $U$ be a compact $C^2-$manifold and suppose $v:U\times (0,T) \to \mathbb R$ is the weak solution of: $\partial_t v= \Delta v+f$ where $f\in L^{\infty}(U\times (0,T))$ I 'm interested in the \$L^...