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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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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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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 ...
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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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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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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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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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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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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{...
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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 ...
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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[...
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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)^\...
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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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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 ...
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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^...
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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 $\Delta$...
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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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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 $\{\phi_{k}...
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$ 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^...
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Question on optimal regularity for the elliptic Neumann problem

I 'm reading a paper at the moment and I have a really hard time understanding the following: Let $U \subset \mathbb R^3$ be open, bounded and connected with a $C^2-$ regular boundary $\partial U$...
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Uniqueness for classical solution of PDE

I have the following conservation law in my hand: $\partial_{t} u + \partial_{x}f(u) = -u$, with the associated initial data $u(x,0) = u_{0}(x)$ - where $u_{0} \in C^{1}$. I have to show ...
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Questions on regularity of Poisson equation

Let $u:\mathbb R^n \to \mathbb R^m$ be a solution of $\Delta u=({\nabla}_u f(u))^T (*)$ for some non-negative function $f:\mathbb R^m \to \mathbb R$. I'm only interested in $n=1,n=2$. I try to ...
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154 views

Boundedness of derivatives of solutions of elliptic differential equations

Consider an elliptic differential equation of the form $$-\nabla.(A\nabla u)=f~~\mbox{ in } \Omega \tag{1}\label{1}$$ with Dirichlet boundary conditions, where $\Omega$ is a bounded open set in $\...
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Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(...
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187 views

How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in C^{\infty}(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \cap L^{\infty}(\...
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53 views

$\Box u= | u |^2 u$ global solution in $C^\infty$

Let $u_0, u_1 \in C^\infty( \mathbb{R},\mathbb{R}^3)$. Consider the cubic defocusing NLW $$(\ast)\begin{cases} \Box u= |u|^2 u \\ (u,\partial_t u) \restriction_{t=0} = (u_0,u_1) ,\end{cases}$$ where $\...
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if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
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156 views

existence and uniqueness of reaction diffusion equation

I am trying to prove uniqueness of equation of the form $$ u_t=u_{xx}+f(u) $$ Can anyone suggest the standard literature for existence and uniqueness for equation like above?
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538 views

Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
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Approximation of an element in the dual of the Sobolev Space

Let $F\in L^2(\Omega)$ be such that $-\text{div}F\in W^{-1,2}(\Omega)$ (Dual of the Sobolev Space $W_0^{1,2}(\Omega)$) be non-negative where $\Omega$ is a bounded domain in $\mathbb{R}^N$. My ...
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Splitting a PDE into two subdomains: Regularity conditions at the interface point(s)

My aim is to split a given problem into two nonoverlapping subdomains. Let us, for example, consider the initial-boundary value problem: Given $u_0 \in L_2(0,1)$ and $f \in L_2(0,1)$, find $u$ such ...
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$u'\in L^{\infty}(0,T;L^{2}(U))\cap L^{2}(0,T;H_{0}^{1}(U))$ regularity in the semigroup approach

Evans' PDE book 2nd, (Theorem 7.1.5) says for parabolic PDEs, if the initial condition is in $H^1_0 \cap H^2$, and the source term satisfies the regularity $\mathbf { f }, \mathbf { f } ^ { \prime } ...
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Example of a function in the weighted Morrey space

Let $1<p<\infty$ and $w\in A_p$ which is the well-known class of Muckenhoupt weights and let $\Omega$ be an open bounded smooth domain in $\mathbb{R}^N$ ($N\geq 2$). For $t>0$ we say that $u$ ...
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52 views

Distributional second-order derivatives of $\frac{e^{-|x|}}{4\pi |x|}$ to show the solution of $u -\Delta u=f$ is in $H^2$

In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)...
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172 views

Gradient Blowup for a Parabolic (Heat) Equation

Let $u(x,t)$ be a solution to the following parabolic PDE: With $\alpha \in (0,1)$, \begin{align} \partial_t u(x,t) &= \alpha (1-t)^{\alpha - 1} \partial_x u(x,t) + \frac{1}{2} \partial_{xx} u(x,...
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46 views

Non piecewise $C^1$ solutions of conservation laws

I studied the following theorem:(Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \rightarrow \mathbb{R} $ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if ...
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37 views

Necessity of data smoothness for abstract parabolic IVP

I have a question regarding conditions on data $f$, $u_0$ to obtain the solution of the parabolic problem. Renardy and Rogers, An Introduction to Partial Differential equations (2nd ed.) states ...
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52 views

Critical spaces in Navier-Stokes theory of well posedness

It is striking that the local in time existence of solution to incompressible 3D Navier-Stokes equations depends only on properties of solutions of the heat equation; such a result holds in more ...
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65 views

Intuition for compact embedding of $H^1([0,1])$ in $L^2([0,1])$?

A compact operator is associated with 'smoothing', i.e. a function will have more regularity after a compact operator has acted upon it. So if we take $\phi \in H^1([0,1])$ and let $I:H^1([0,1]) \to ...
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34 views

1-D Heat Equation, bounding difference in $\alpha$ given surface temperature

Given the following heat equation and boundary conditions: $u_t= \alpha(x) u_{xx}, \quad x \geqslant 0 \,\,\text{and}\,\, \exists b \geqslant a > 0, a \leqslant \alpha(x) \leqslant b\\ u_x(0,t)...
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47 views

Faedo-Galerkin method for the 1-d von-karman system

I want to prove the existence of a solution to the following system $$\eqalign{ & {u_{tt}}-{u_{ttxx}}+{u_{xxxx}}-{({u_x}({v_x} + \frac{1}{2}u_x^2))_x}=0 \cr & {v_{tt}}- {({v_x} + \frac{...
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75 views

regularity of Harmonic Oscillator

This problem is from $Spectral \ Theory \ and \ its \ Applications,\ Bernard\ Helffer, \ Page \ 37$. It says that if $u \in H^1(\mathbf{R^m}), x_j u \in L^2(\mathbf{R^m}),(-\Delta+\vert x \vert^2+1 )...
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77 views

Does the continuity of r.h.s in elliptic PDE imply C^2 solution?

my problem is concerning an elliptic PDE $$ Lu + \lambda u = f(\cdot,u,\nabla u), $$ $$ \frac{\partial u}{\partial n}=0$$ in a Lipschitz domain $\Omega \subset \mathbb{R}^d$ and $n$ is the outward ...
2
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0answers
135 views

If $u\in H_0^1(\Omega)$ satisfies $-\Delta u=f$, with $f\in C^{\alpha}(\bar{\Omega})$, then $u\in C^{2,\alpha}(\bar{\Omega})$.

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with smooth boundary, $n\geq2$. Schauder regularity theory says: If $u\in H_0^1(\Omega)$ satisfies $-\Delta u=f$, with $f\in C^{\alpha}(\bar{\...
2
votes
0answers
31 views

Description of strongly elliptic functions

Is there a way to describe the set of strongly elliptic functions $$ A:=\{a:\mathbb{R}^n\to\mathbb{R}^n\mid\forall x,y\in \mathbb{R}^n,\ (a(x)-a(y))\cdot(x-y)\ge K\|x-y\|^2 \}$$ maybe in terms of ...
2
votes
0answers
104 views

Understanding regularity of functions and how they are affected by increasing exponents?

Take the function $f_1(x) = x^{0.5}$. This function is continuous in $\mathbb{R}$ but its derivative $f_1'(x) = 0.5 x^{-0.5}$ is not. If we then consider $f_2(x) = x^{0.9}$, this function is also ...
2
votes
0answers
43 views

Elliptic estimates on a compact domain

Let $\Omega \subset \mathbb{R}^n$ be a ball with boundary. Consider the operator $-\Delta + V$ on $\Omega$, where $\Delta$ is the Dirichlet Laplacian and $V$ is a positive smooth potential. Let $u$ ...
2
votes
0answers
67 views

Well-posedness of non linear evolution PDE

I have the following evolution PDE problem \begin{align} \frac{\partial f(x,t)}{\partial t} &= \frac{1}{2}\int_{y\leq x,x \in I} Q(x-y , y) f(x-y,t) f(y,t) dy - f(x,t) \int_{I} Q(x, y) f(y,t) ...
2
votes
0answers
35 views

How does the nonlinearity growth affect the regularity of solutions?

Consider equation \begin{equation*} \begin{cases} -\Delta u=f(u)&\text{ in }\Omega,\\ u=0 &\text{ on }\partial\Omega, \end{cases} \end{equation*} where $|f(u)|\leqslant C_1+C_2|u|^{...
2
votes
0answers
78 views

Reference for existence of strong solution for semilinear parabolic PDE on unbounded domain with monotone source

I am considering the existence of solutions for the following semilinear parabolic PDE \begin{cases} u_t+Lu=f(t,x,u,u_x), &(t,x)\in [0,T]\times\mathbb{R},\\ u(0,x)=g(x), \end{cases} where $L$ is a ...
2
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0answers
139 views

Conservation Laws: Difference and Reasonability Weak solutions, Integral solutions, distributional solutions

Suppose we have for $T\in\mathbb{R}_{>0}$ a conservation law of the following general form \begin{align} \dot{u}(t,x)+(a(t,x,u(t,x)))_{x}&=0 && (t,x)\in(0,T)\times\mathbb{R}\\ u(0,x)&...