Questions tagged [regularity-theory-of-pdes]
This tag is for questions concerning the smoothness of weak solutions to partial differential equations.
775
questions
113
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9
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Why should I "believe in" weak solutions to PDEs?
This is a sort of soft-question to which I can't find any satisfactory answer. At heart, I feel I have some need for a robust and well-motivated formalism in mathematics, and my work in geometry ...
- 17.9k
24
votes
2
answers
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viscosity solution vs. weak solution
viscosity solution vs. weak solution
I am confused between the two. Is one a subset of the other or they are the same/completely different notions? Suppose I have an equation, $u_t=\mathcal{L}u$ for ...
- 1,179
18
votes
1
answer
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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 ...
user230715
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votes
2
answers
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cutoff function vs mollifiers
$\boldsymbol{Q_1}$ What are cutoff functions? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions and how they differ from mollifiers?...
- 3,065
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votes
2
answers
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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 ...
- 23.9k
16
votes
0
answers
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Regularity of parabolic PDEs for large $\lambda$
Let $\Omega$ be a sufficiently smooth domain, $T>0$, and $L$ be the following elliptic operator of the divergence form:
$$Lu(t,x)=a^{ij}(t,x)u_{ij}(t,x),$$
such that $a^{ij}\in C([0,T]\times\bar{\...
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votes
2
answers
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What is elliptic bootstrapping?
While reading about elliptic differential operators, I have seen the phrase elliptic bootstrapping in several places, but none of them explain exactly what it means. I know it has something to do with ...
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10
votes
1
answer
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Ricci Flow: PDE details?
Over the past few weeks I have been reading 'Ricci flow: An introduction' (Chow and Knopf) which is, in my opinion, a very well written and quick introduction to the topic. However I find that the ...
- 612
9
votes
1
answer
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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 ...
- 3,065
8
votes
2
answers
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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 ...
- 3,065
8
votes
1
answer
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Is the composition of a Sobolev function and a smooth function Sobolev?
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f \in W^{1,p}(\Omega)$ is continuous*, and $g \in C^{\infty}(\mathbb{R})$.
Is it true that $g \...
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votes
1
answer
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Reference complementing Hairer's "Introduction to Stochastic PDEs"
I want to study stochastic partial differential equations, and the standard reference seems to be Hairer's notes - after all, he got the first Fields Medal for work on this topic.
However, I also ...
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7
votes
2
answers
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Sobolev Embedding into $L^{\infty}$
I tried to find the question here, but I couldn't.
I'm a bit puzzled by Sobolev embeddings at the moment.
In my lecture notes, I found the statement
"If $\Omega \subseteq \mathbb{R}^d$ is ...
- 168
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votes
1
answer
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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 ...
- 3,065
7
votes
1
answer
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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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7
votes
1
answer
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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.
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votes
1
answer
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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 ...
- 3,917
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votes
2
answers
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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 ...
- 1,094
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0
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Community-compiled errata for Bertozzi and Majda's "Vorticity and Incompressible Flow"?
While this is a great book, I think there isn't an errata for Bertozzi and Majda's "Vorticity and Incompressible Flow", so can we compile a list of errors here? Since this book is a classic reference ...
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7
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0
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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 ...
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2
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In what conditions a weak solution is a classical solution?
I'm studying elliptic equations in divergence form
$$-D_{j}(a_{ij}(x)D_{i}u) + c(x)u = f(x) \text { in a domain } \Omega \subset \mathbb{R}^{n}$$
I call a function $u \in H^{1}(\Omega)$ a weak ...
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1
answer
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Are distributional (local) solutions to the heat equation smooth?
I have thought about this apparently simple problem.
Question:
Let $\Omega\subset \mathbb{R}_{x,y}^2$ be an open subset. Let's suppose we have $u\in\mathscr{D}'(\Omega)$ that satisfies
$$ (\partial_y-...
- 4,182
6
votes
2
answers
638
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Energy estimate for heat equation
Consider the following heat equation in a bounded smooth domain $D \subset \mathbb{R}^d$:
\begin{align*}
u_t -\triangle u &= f, \qquad (x,t) \in D \times (0,T)\\
\partial_n u &=0, \qquad (x,t) ...
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6
votes
1
answer
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Financial importance of the diffusion term in Black-Scholes partial differential equation
Consider the Black-Scholes equation $$\begin{equation}\label{eq3}
\frac{\partial{V}}{\partial{t}}+\frac{1}{2}\sigma^2S^2\frac{\partial^2{V}}{\partial{S}^2}+(r-D)S \frac{\partial{V}}{\partial{S}}-rV=0,...
- 2,174
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votes
1
answer
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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 $\det df>0$...
- 25.1k
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votes
3
answers
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Bounding $L^{\infty}$ norm for elliptic PDE
I'm trying to prove the following: If $L$ is a uniformly elliptic PDE, without costant or linear terms and with essentialy bounded coefficents $a^{ij}$. If $u\in H^1(B_2)$ satisfies $Lu = f$, then
$$|...
- 761
6
votes
1
answer
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Loss of derivatives
In many books on pdes the expression "loss of derivatives" is used when some estimates on solution are proved.
Can someone clarify to me (maybe with an example) the meaning of this ...
- 61
6
votes
0
answers
237
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Gradient estimate of degenerate parabolic equation
Suppose $u : [0,1] \times [0,T] \to \mathbb{R}$, we consider the problem
\begin{equation}
\left\{\hspace{5pt}\begin{aligned}
&\dfrac{\partial u}{\partial t} - a(x,t)x^2\partial_x^2 u - b(x,t)x\...
- 465
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votes
0
answers
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Orthonormal basis of the Schwartz space
The Hermite functions $(h_n)_{n \ge 0}$ are the eigenfunctions of $T f = (x+\partial_x)(x-\partial_x) f$, they are defined by their generating function $g(x,t)=\sum_{n=0}^\infty h_n(x) t^n = e^{-x^2/2+...
- 78k
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0
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Is being in the Sobolev space of power $\frac{d}{2}$ necessary for having well defined point evaluations?
From the Sobolev embedding theorem we know that for $\alpha = \frac{d}{2}$, $W^{\alpha, 2}(\mathbb{R}^d)$ is continuously embedded in $C^0(\mathbb{R}^d)$. Especially the point evaluations are in the ...
- 116
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0
answers
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Elliptic PDE: dependence of estimates on ellipticity constants?
Consider an elliptic PDE such as
$$\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 ...
user360874
6
votes
0
answers
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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{...
- 3,801
6
votes
0
answers
792
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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 ...
- 899
5
votes
2
answers
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$\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 ...
- 433
5
votes
3
answers
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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.t.\...
- 433
5
votes
1
answer
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Heat equation with dirac delta as source term
I have the following equation in $\mathbb{R}^2$,
$ \partial_t u(x,t) - \Delta u(x,t) = \delta_0, $
with $u(x,0) = 0.$
I received this equation and I instantly got suspicious. I have tried to prove ...
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1
answer
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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^...
- 2,096
5
votes
1
answer
527
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Boundedness of the solutions of PDEs
Consider the solution of the following heat equation in $\mathbb{R}^n\times (0,T)$:
$$\left\{\begin{array}{rcl}
\partial_tu-\Delta u + c(x) u&=& 0\\
u(x,0)&=& f
\end{array} \right.$$
...
user712236
5
votes
1
answer
497
views
$BV(\Omega)$ is embedded compactly in $L^1 _{\mathrm{loc}} (\Omega)$
Definition:
We say that a function $u: \Omega \rightarrow \mathbb{R}$ is a function of bounded variation iff $u\in L^1(\Omega)$ and $\sup\left\{\int\limits_{\Omega}u \operatorname{div}\phi : \phi \in ...
- 1,015
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votes
1
answer
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What is regularity of solution of Dirichlet problem with Dirac distribution as boundary data?
I was thinking if we have Dirac distribution as boundary condition, then what will be regularity of solution. Problem is following,
$$
\left\{\begin{matrix}
\nabla_x(\gamma(x)\nabla_x u(x,y))=0 & ...
- 6,593
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votes
2
answers
317
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$W_{loc}^{1,2}$ regularity of nonnegative subharmonic functions
I'm trying to solve the following excercise:
Let $v \in C(D)$ be a nonnegative subharmonic function in an open set $D$. Prove that $v \in W_{loc}^{1,2}(D)$
The problem has the following hint:
...
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1
answer
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Equivalence between norms in $H_0^1(\Omega)\cap H^2(\Omega)$.
Based in many questions and answers like [1, 2, 3 ] and a comment a good comment here [4].
I would like to know that the space $H=H_0^1(\Omega)\cap H^2(\Omega)$ can be equiped with this norm
$$\tag{...
- 303
5
votes
1
answer
442
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Doubt about the proof of Moser Iteration in Gilbarg & Trudinger's book
I was reading Theorem 8.15 about Moser Iteration in Gilbarg and Trudinger's monograph. I understand all the steps of the given proof, but I have the following doubts which could not be cleared by a ...
- 6,593
5
votes
1
answer
225
views
Difference quotient for Hölder continuous functions
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in C^\alpha_{\mathrm{loc}}(\Omega)$. For $h>0$, $1\leq k\leq n$, let $$D_k^hu(x)=\frac{u(x+he_k)-u(x)}{h}$$ where $e_k$ is the $k$-th ...
- 8,556
5
votes
1
answer
571
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:
$$
\frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{...
- 441
5
votes
0
answers
87
views
PDE with a non-classical boundary condition
Assume that one has a classical PDE, say: $u_t(t,x)-u_{xx}(t,x)=0$, $t\in (0,1)$, $x\in (0,2)$, and $u(0,x)=0$. Then we can prove existence (and uniqueness) of solution when boundary conditions: $u(t,...
- 271
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votes
0
answers
390
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Distributional and weak solutions
Consider a differential equation,
$$L(T)=f,$$
where $L$ is a linear ordinary differential operator with smooth coefficients and $f$ is a $L_{loc}^1$ function.
Does there always exist a distribution T ...
- 1,830
5
votes
0
answers
126
views
Elliptic regularity of harmonic forms in $L^1$
$\newcommand{\M}{M}$
Let $\M$ be a smooth oriented Riemannian manifold.
Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if
$$
\...
- 25.1k
5
votes
0
answers
32
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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 ...
- 25.1k
5
votes
0
answers
113
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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)^\...
- 742