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Questions tagged [regularity-theory-of-pdes]

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

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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 ...
7
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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 ...
7
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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 ...
6
votes
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.
6
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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 $\det df>0$...
6
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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 ...
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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 ...
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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 ...
5
votes
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 ...
5
votes
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 ...
5
votes
1answer
289 views

Need source for elliptic regularity on unbounded domains

I need a source that provides a $W^{2,2}$-regularity result for linear elliptic systems on unbounded domains. To be more specific, I study the equation $$ -\Delta w - ic \partial_1 w + \left( \frac{c^...
5
votes
1answer
748 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
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0answers
76 views

Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ Let $\M$ be a smooth oriented Riemannian manifold. Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if $$ \...
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0answers
94 views

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 ...
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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 ...
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0answers
86 views

Curvature of the boundary vs. normal derivative of the first eigenfunction

Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$. That is, $$ \left\{ \begin{aligned} -\Delta \varphi_1 &= \lambda_1 \varphi_1 &&...
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0answers
81 views

Regularity of quasilinear PDE up to a smooth part of the boundary

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain. Consider the Dirichlet problem $$ -\Delta_p u = |u|^{q-2} u \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega, $$ where $...
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0answers
111 views

Solution to Singular Free Boundary PDE

As part of my research, I have come across the following problem and I am trying to tackle it. Let $(X_t)_{0 \leq t \leq T}$ be a mean controlled Brownian Motion with the following dynamics \begin{...
5
votes
1answer
227 views

Continuity of PDE solutions with respect to coefficients

Suppose I have a PDE, for example the Fokker-Planck one, in which I am mostly interested: $$ \frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{...
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0answers
74 views

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 ...
4
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3answers
279 views

Weak solutions to $\Delta u=f$ are in $W^{2,2}$

I believe the following statement is true. Let $\Omega$ be a smoothly, bounded domain in $\mathbb{R}^{n}$. The statement: Let $u\in H^{1}(\Omega)$ so that there exists $f\in L^{2}(\Omega) \;s....
4
votes
2answers
737 views

Proving the range of operator is closed

I have a hard time understand (2) of the Fredholm alternative in Evan's Appendix. To prove the image of $I-K$ is closed, what result from functional analysis is used? I am lost in understand the ...
4
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1answer
2k views

Density of smooth compactly supported functions in Sobolev space over unbounded domain.

Prove that $C^{\infty}_ {c}(\mathbb{R}^n)$ is dense in $W^{k,p}(U)$ for any open $U\subset \mathbb{R}^n$ with $\partial U\in C^1.$ In which $p\in [1,\infty)$ Note: In Lawrence Evans's PDE text, the ...
4
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1answer
86 views

the proof of variational principal for the principal eigenvalue (checking orthonormal subset)

Hi I am looking at part 3 of the proof in Evans Chapter 6. I have difficulty understanding "Furthermore from (6) and (7) we see that $(\lambda_k^{-1/2} w_k)$ is an orthonormal subset of $H_0^1(U)$. ...
4
votes
1answer
308 views

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

Consider the following boundary value problem (BVP) $$\matrix{ {{\Delta ^2}H = 0,} \hfill & {} \hfill & {{\rm{in}}\,} \hfill & \Omega \hfill \cr {\partial _y^2H = 0} \hfill & ...
4
votes
1answer
145 views

Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with $...
4
votes
1answer
48 views

Higher differentiability of weak solutions to 2nd order elliptic PDEs with mixed boundary conditions

I am interested in regularity results for 2nd order elliptic PDEs with mixed boundary conditions like $$\left\{\begin{array}{rl}-\text{div}(a\nabla u) =& f &\text{in }\Omega, \\ u=&\...
4
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0answers
56 views

Is the following PDE an ill-posed problem?

I want to solve the following PDE: $$\frac{\partial h}{\partial u}=\frac{1}{2}\frac{\partial^{2} h}{\partial y^{2}}+\mathbf{1}_{\{u\leq v-t\}}yh-\beta\mathbf{1}_{\{u\leq v-t\}}h$$ such that $(u,y)\in[...
4
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0answers
50 views

Smoothness of solution of parabolic type PDE

since I am not very familiar with this area, I have some queries(basic) about the regularity of solutions of a parabolic type PDE. Suppose we have the following differential operator: $Lu(x)=b(x)^\...
4
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0answers
52 views

Regularity for the harmonic equation in coordinates

$\newcommand{\lap}{\Delta}$ $\newcommand{\al}{\alpha}$ $\newcommand{\be}{\beta}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\pl}{\partial}$ Let $(M,g),(N,\eta)$ be smooth ...
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0answers
64 views

Elliptic PDE: dependence of estimates on ellipticity constants?

Consider an elliptic PDE such as $$\mbox{tr}(A(x)D^2 u(x)) = f(x), \quad x \in B_1\subset\mathbb R^n,$$ where $A(x)$ is symmetric and $0<\lambda I \leq A(x) \leq \Lambda I$ for all $x$. Let us ...
4
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0answers
291 views

Poisson Equation - $L^2$ boundary regularity

Let $\mathbb{R}_{+}^n = \{(x',x_n): x' \in \mathbb{R}^{n-1}, x_n > 0 \}$ be the half space. It is known that if $u \in H_0^1(\mathbb{R}_{+}^n)$ is a weak solution to $-\Delta u = f$ with $f \in L^...
4
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1answer
59 views

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'...
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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 $\Delta$...
4
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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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0answers
63 views

I want to proof that the lebesgue measure of the set below is positiva. Help me!

Let $\Omega$ be a domain limited with smooth boundary in $\mathbb{R}^{n}$ and consider the Sobolev space $H^{1}_{0}(\Omega)$ equiped with the norm $||u||=\int_{\Omega}|\nabla u|^{2}dx$. Let $\{\phi_{k}...
4
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1answer
293 views

Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume $$a^{...
3
votes
1answer
46 views

Does a continuous function in $W^{1,p}$ for $p<n$ lie in $W^{1,n}$?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain. Suppose that $f \in W^{1,p}(\Omega)$ for some specific $1<p<n$, and that $f$ is continuous. Is it true that $f \in W^{1,n}_{loc}(\...
3
votes
1answer
79 views

PDE without finite time blow up for small initial data?

Is there a pde (or a class of pde), for which, having small initial data, is a necessary and sufficient condition for its solution to not have a finite time blow up?
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2answers
133 views

How to solve this second order linear pde?

I have the following pde for $f(t,x,y)$: $a x^2 f_{xx} + bxf_x + f_t - bxy + c = 0$ subject to $f(T,x,y)=0$ for all positive $x,y$, where $a,b$ and $c$ are constants. The equation seems to be ...
3
votes
2answers
66 views

Non-existence of weak solution in one dimension

Let $\Omega=(1,\infty).$ Then for any given $f\in L^2(\Omega),$ the equation $$ -u''=f\,\,\text{in}\,\,\Omega, $$ does not admit any weak solution in $W_{0}^{1,2}(\Omega).$ I tried the solution by ...
3
votes
1answer
421 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0 & \text{on } \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}...
3
votes
1answer
88 views

A regularity question for elliptic PDE

I revise the question according to the comment of Andrew Let $D$ be an elliptic differential operator with analytic coefficient on $C^{\infty}(\mathbb{R}^2)$, the space of complex valued ...
3
votes
1answer
97 views

Questions on Nonlinear Elliptic Theory by Schauder

I recently started to study about Elliptic theory and below is a brief introduction my professor made: Let $\;u:\mathbb R^n \to \mathbb R\;$ and $\;f:\mathbb R \to \mathbb R\;$ two functions ...
3
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1answer
267 views

Distributional Derivative of delta function.

I am given with Heaviside function $$H(x) = \begin{cases} 0 & if \ \ x \in (-\infty, 0) \ \\ 1 & if \ \ x \in (0, \infty) \end{cases}$$ Now I have calculated its ...
3
votes
1answer
1k views

Uniform convergence in the proof of properties of mollifier (Evan's approach)

I am still trying to understand Evans' proof on the properties of mollifier. In the proof of (iii), I understand that the crux of the proof is that uniform continuous function on compact set is ...
3
votes
1answer
137 views

Regularity of Parabolic pde

In Evans' pde Book, In Theorem 5, p. 360 (old edition) which concern regularity of parabolic pdes. he consider the case where the coefficients $a_{ij},b_i,c$ of the uniformly parabolic operator (...
3
votes
1answer
51 views

Proving some norm inequalities

Let $u \in H^{m}(U)$ and $W \subset\subset U$ that is $W \subset \bar{W} \subset U$ with $\bar{W}$ is compact in $U$ and $W$ is compactly contained in $U$. Is this true that $$||D^{\alpha}u||_{L^{2}(...
3
votes
1answer
192 views

Is $H^2(\Omega)\cap H_0^1(\Omega)$ compactly embedded on $H_0^1(\Omega)$?

Considering $\Omega$ bounded and $\partial \Omega$ smooth. I already know that $H^2(\Omega)\cap H_0^1(\Omega)$ is continuously embedded on $H_0^1(\Omega)$, thus if I take a bounded sequence in $H^2(\...
3
votes
0answers
40 views

$ 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^...