Questions tagged [regularity-theory-of-pdes]

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

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35 views

If $u \in H_0^1(\Omega) \cap L^p(\Omega)$ and $\Delta u \in L^p(\Omega)$ then $u \in W^{2, p}(\Omega)$

How to show the following? If $u \in H_0^1(\Omega) \cap L^p(\Omega)$, $\Delta u \in L^p(\Omega)$ then $u \in W^{2, p}(\Omega)$. This is part of the Brezis-Kato regularity argument as presented by ...
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$H^1$-conforming approximation for elliptic PDE with discontinuous coefficient?

everyone Suppose I want to solve the diffusion equation $$-\nabla\cdot a \nabla u=f, \\u=0 \text{ on } \partial \Omega,$$ $f \in L^2(\Omega)$, $\partial \Omega$ is smooth. I use the standard node-...
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Fundamental Solution of the Heat Equation on an Arbitrary Domain

It is well known (cf. Evans §2.3.1) that $$\Phi(x,t):= \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x|^2}{4t}}$$ is the fundamental solution of the heat equation on $\mathbb{R}^n$ for $t>0$, i.e., $$\begin{...
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Parabolic Bootstrapping

I started studying parabolic pdes. Often I come across an integral solution where the regularity is proven by a standard bootstrap argument or by standard parabolic results, but it is never explained ...
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Brezis-Kato regularity argument - Some questions about Struwe's proof Part II

The following is in Appendix B of Struwe's Variational Methods Let $u$ be a solution of $-\Delta u = g(x, u(x))$ in a domain $\Omega \subset \mathbb R^N$, $N \geq 3$, where $g$ is a Carathéodory ...
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77 views

Brezis-Kato regularity argument - Some questions about Struwe's proof

The following is in Appendix B of Struwe's Variational Methods Let $u$ be a solution of $-\Delta u = g(x, u(x))$ in a domain $\Omega \subset \mathbb R^N$, $N \geq 3$, where $g$ is a Carathéodory ...
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132 views

Regularity for a Nonhomogeneous Heat Equation

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with smooth boundary, and let $T>0$. Consider the non-homogeneous heat equation with Dirichlet boundary condition $$\begin{aligned} u_t - \...
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Euristic and intuitive idea behind the theory of viscosity solutions

As the title suggests, I am kinda struggling to understand the basic idea behind viscosity solutions theory. The theorems are a lot different from what I saw in classical theory for PDEs (with Sobolev,...
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weak star convergence and time derivative in Bochner spaces

Assume that for a sequence of vector valued functions $u(\varepsilon):(0,T)\to L^2(\Omega)^3$ we have $\frac{1}{2}(\partial_i u_j(\varepsilon)+\partial_j u_i(\varepsilon))\stackrel{*}{\rightarrow}e_{...
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A regularity result in Bochner spaces, simmetry of second derivatives (time and space)

For a function $u:(0,T)\to L^2(\Omega)$, assume that we have shown that $\frac{\partial u}{\partial x_i}=\partial_i u \in L^\infty(0,T;L^2(\Omega))$ and also that $\partial_t u \in L^\infty(0,T;L^2(\...
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Continuity of weak solutions to wave equation with time-dependent coefficients

Consider the following second-order wave equation $$ u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega $$ with boundary conditions $$ u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
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Moser Iteration for Laplacian with Hardy potential

I am reading the following proof of Cao-Yan's 2010 CVPDE Paper. It's a property for solutions of Laplace equation with Hardy potential $|x|^{-2}.$ The space dimension $N \geq 3.$ Their proof is in the ...
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37 views

Harnack inequality for viscosity solutions of $F(D^{2}u,Du,x)=f$

I would like to know if there is an harnack inequality for the viscosity solutions of the following pde in the literature: $F(D^{2}u,Du,x)=f \in L^{\infty}(B_{1})$, for a function $u \in C(B_{1})$ and ...
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Solvability of Poisson equation with Cauchy boundary condition

I am interested in any explanation/comment/reference you could provide me regarding the solvability of the Poisson problem with Cauchy boundary data $$ \begin{cases} -\Delta u = f \ &\textrm{in}\ ...
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General solution of a PDE

I am trying to solve for x>0 and y>0 the following PDE: $$ x^2 u_{xx} -y^2 u_{yy} +x u_{x} - y u_{y} =0 $$ The characteristics are $$ \frac{dy}{dx} = \pm \frac{y}{x} $$ so I get $\xi = \frac{y}{...
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Is there any good reference for Second order Elliptic PDE other than Gilberg Trudinger?

I had done a course in PDE. I wanted to learn Elliptic PDE on my own. My teacher suggested me book Gilberg Trudinger. I am reading that. It is a good book. I am understanding almost all calculation. ...
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Poisson equation incompatible data regularity

Let's have a function $u=x^3 + y^3$ defined on a smooth and convex domain $\bar{\Omega}$. The same function is $u\in C^\infty$ and is a classical solution to: $$\Delta u = 6x + 6y\, \text{ on } \...
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$V$ is Compactly Embbeded in it self?

Can we say that $V$ is compactly embbeded in it self $$V=\left\{u\in H^{1}\left(\Omega\right):\gamma_{0}\left(u\right)=0 \text{ on } \Gamma_{0}\right\}$$ I would like to use aubin-lions theorem.
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$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation

We consider a Fokker-Planck equation: $$ \partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0, $$ with initial ...
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Uniform convexity and embedding result in the Heisenberg group

Can somebody please help me with compact embedding result and uniform convexity property of the Sobolev space $W^{1,p}(\Omega)$ where $\Omega$ is a bounded smooth domain in the Heisenberg group? ...
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Regulartiy of weak solution to second-order parabolic equation (Evans)

I have a problem with the following part of Evans book PDE. It is in the proof of the improved regularity of weak solution to a second order parabolic equation (Theorem 5, Chapter 7.1, page 361-364). ...
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48 views

weak solution / variational calculus

We have a one dimensional boundary value problem $$ - (\omega u_x)_x = f \, \text{for} \, -1<x<1 \\ u(-1)=u(1) = 0 $$ with $$ \omega(x): = \sqrt{(1-x^2)} \, \text{and} \, f(x) := x \, \text{for} ...
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weak solution/variational method and the Lax Milgram Theorem

I am trying to understand the variational method and the connection to the Lax-Milgram-Theorm. I don't know how to use the theory to solve this exercise. Let $\epsilon > 0$ and we have the ...
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Are both sides of a distributional equation in the same space?

I'm trying to understand a paper, but will try to hold this question more general. (So if you need more information i can state the "real" problem.) Say I have $$u \in L^2\\h\in W^{-1,2}$$ and the ...
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Parabolic to Elliptic

Let $u:\Omega\times(0,\infty)\to(0,\infty)$ satisfies the heat equation $$ u_t=\Delta u, $$ then does the function $v(x)=u(\cdot,t)$ defined for every fixed $t$ satisfy the equation $$ \Delta u=0 \...
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39 views

Need help in understanding an argument.

Suppose $u: A\times B\to (0,\infty)$ be a function such that $$ \int_{x\in A}\int_{y\in B}u^2(x,t)\,dx dt\leq \int_{x\in A}\int_{y\in B}f(u(x,t))\,dx dt. $$ Now let us fix $\hat{t}\in B$, and define $...
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$\Delta u = u$ implies $u$ infinitely differentiable, without Sobolev techniques

Suppose $u:\mathbb{R}^n \to \mathbb{R}$ is a $C^2$ function which satisfies $\Delta u = u$. Can we prove that $u$ is $C^{\infty}$ without using the elliptic regularity theorem or any machinery of ...
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Bootstrapping For Elliptic Operators in Holder Space

Do we have a proposition in the following form: L is an elliptic operator on $\Omega$. If we have $Lu = v$ with $v\in C^{k-2,\beta}(\Omega)$ and $u\in C^{k}(\Omega)$. We actually know that $u\in C^{k,...
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1answer
32 views

Order of accuracy for non-smooth solutions and non-smooth local truncation errors

I'm working with numerical methods for solving PDEs (Linear Advection/Euler equations with temporal and spatial discretisation) using finite difference/finite volume methods. In these simulations I ...
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41 views

Infinite differentiability in the interior, regularity in PDE Evans

I am confused about the proof of the following theorem which is found in Evans' PDEs, Chapter 6.3. Theorem 3(the infinite differentiability in the interior). Assume $$ a^{ij},b^i,c\in C^\infty(U)...
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An inequality in regularity result for a PDE

My question arise in the proof of the following theorem but we don't need to understand the theorem because my question is just one inequality involved in the proof of it. But let me state the theorem ...
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1answer
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What is the meaning of formal calculation in PDE?

I am a math major and I am studying PDE. When I read papers in this field, I sometimes find some formal calculations, for example, they differentiate as if the function is smooth or they interchange ...
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Why is this function a local minimum of this integral functional? (related to elliptic PDEs)

This is actually a step in a bigger proof about regualarity properties of elliptic systems, but I'll write only what I don't understand. Let $L: \mathbb{R}^{m \times n} \to \mathbb{R}$ be $C^2$ and ...
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Passage to limit in differential equation

we consider the equation \begin{equation} \partial_t u (t , x) + v (t, x) \cdot \nabla u ( t, x) = \kappa \Delta u (t , x) + F ( t, x, u (t , x) ) \qquad \mbox{in} \ \ \mathbb{R}_+ \times \mathbb{R}^...
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Condition for global existence of solutions of PDEs when local existence is given

I have a system of non-linear PDEs for which I can prove there exists a unique solution at each point $(x,t)$ of my domain $\Omega\times(0,\infty)$, where $\Omega\subset{R}^n$ is regular and bounded. ...
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Bootstrap argument in De Giorgi's regularity theorem for elliptic PDEs

I was studying De Giorgi's regularity theorem for elliptic systems, and there is something I don't quite understand. The main part of the theorem is dedicated to the proof that certain functions (that ...
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Non-homogeneous transport equation with Hölder continuous coefficients

I'm working with an non-homogeneous transport equation with Hölder Continuous coefficient. I'm seeking for some reference in this subject. Precisely, I'm concerned with the following problem: \begin{...
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1answer
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Deep Learning Algorithm Global Minimum

I am struggling with a problem of calculating the global minima of Neural Networks in Natural Language Processing. The first method I used is to finde the global minimum based on convexity prperties. ...
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Getting $|Du(x)-Du(y)|\leq C|x-y|$?

I'm reading materials about Mather theory, weak KAM theory. Well, the question is simple; Assume that $$|u(x+h)-u(x)-Du(x)\cdot h|\leq C|h|^2$$ for all $x\in\mathcal{M}_0,\ h\in\mathbb{R}^n$. The ...
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Characterization of space $D(A^2)$ in semigroup theory for laplacian operator

Let us assume $\Omega$ is sufficient smooth. Let $H=L^2(\Omega)$ and define $A:D(A)\subset H\to H$ by $Au=\Delta u$, with $D(A)=H_0^1(\Omega)\cap H^2(\Omega).$ Brezis's book, Brezis, Functional ...
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Application of Brezis ieb lemma

I am facing difficulty to understand the argument in the following article:https://projecteuclid.org/download/pdf_1/euclid.ade/1355867973 on page 205 in line 6, how does the argument $J_u'(1)+c^2=0$ ...
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No regularity requirements when the continuity equation is in divernce form

I dont understand the Remark 2 in this Text by Ambrosio and Crippa (see below a screenshot). How does the divergence form relate to no regularity requirements? The part in question
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Extension of solution PDE

Let us consider a non-negative function $u \in C^{0,\alpha}(B_1)$ such that $\Delta u=1$ in the set $\{u>0\}\cap B_1$. Is it true that $$ \Delta u = \chi_{\{u>0\}} \quad\mbox{in }B_1? $$ In ...
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Higher Sobolev regularity for Elliptic equation

Let $\Omega$ be a smooth subset of $R^d$. Suppose that $u\in W^{2,2}(\Omega)$ solves $$\text{trace}(AD^2u)=a_{ij}\partial_{ij}u=F,$$ in the weak sense, where $A=(a_{ij})_{ij}$ is a uniformly elliptic ...
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An estimate of an integral of a continuous and bounded function over time and space

Let $r>0$ and $f:\mathbb{R}\times(-r,0)$ be a bounded and continuous function. Then there exists $\tau\in(-r^{2},0)$ and $\epsilon_0>0$ with $J:=[\tau-\epsilon_0 r^2,\tau]\subset(-r^2,0)$ such ...
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43 views

Generalization of the Pucci extremal operators.?

Let $S$ the set of symmetric matrices, and $S_{\lambda,\Lambda}$ the set of symmetric matrices whose eingenvalues belong to $[\lambda,\Lambda]$, we define the Pucci's extremal operators as $\mathcal{...
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99 views

Interior $H^2$ regularity (proof)

The problem is related to the proof of the interior $H^2$ regularity theorem from Evans's book Partial Differential Equations (sec 6.3). In the proof, I am having difficulty in understanding how the ...
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20 views

Heat equation and convergence of approximate solution

We consider the equation \begin{equation} \partial_t u(t,x)+ v(t,x) \cdot \nabla u(t,x)= \kappa \Delta u(t,x) + F(t,x,u(t,x)) \ \mbox{ dans } \mathbb{R}_+ \times \mathbb{R}^d,....(1) \end{equation} ...
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9 views

What does the Linearised Version of a System of PDEs tell you about the Non-linear System?

I am trying to learn some basic PDE theory so apologies if this is a very naïve question. I am considering a paper by Sirakov on estimates for weakly coupled systems of elliptic PDE and it is stated ...
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24 views

Higher interior regularity for second order elliptic equation

Higher Interior regularity of second order elliptic equation i am not underderstand how the inequality (37) comes from (29)-(32) and (36), and how $\tilde{f}\in L^2(W)$ (?) $||\tilde{f}||_{L^2(W)}...

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