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regularity of solutions of PDEs.

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Elliptic regularity theory in $\mathbb{R}^2$

I recently encountered two papers discussing elliptic PDEs and variational methods. The first paper claims that according to regularity theory, the solution to $-\Delta u = ug(u^2)$ in $\mathbb{R}^2$ ...
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Modulus of Continuity, Heat Flow, and Derivative Estimates

Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by \begin{align} (P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right], \end{align} where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
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Techniques to estimate PDE which are elliptic in some directions and degenerate in others

I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
Aidan Backus's user avatar
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Continuity of the constant in maximal Sobolev regularity

Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
Keba's user avatar
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A question about regularity results in the Elliptic case which are given by Schauder theory

I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
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About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain

I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$: Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
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Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it. We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
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Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
Akira's user avatar
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Parabolic regularity for weak solution with $L^2$ data

I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions: $$\begin{cases}\...
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Properties of "potential vector field" in Helmholtz decomposition

It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as $$ F= \nabla V+ \nabla \times R$$ with $V$ a potential and $R$ another vector field. These components ...
tommy1996q's user avatar
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1 answer
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Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$

I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given: \begin{align*} \partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\ ...
Paul Joh's user avatar
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Smoothness of solutions to wave equation in a bounded domain

Consider the wave equation \begin{equation} \partial_t^2 u - \sum \partial^2_{x_i} u =0 \end{equation} in a bounded domain $M$ with $C^\infty$ boundary, and the boundary conditons \begin{equation} u(...
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How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?

Already asked in SE but no response, I think it also reasonably belongs here. https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions Basically what the title says, plus ...
Clara Torres-Latorre's user avatar
2 votes
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Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
alia's user avatar
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1 vote
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120 views

Regularity of elliptic equation with Neumann boundary conditions

In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
Cathelion's user avatar
2 votes
0 answers
176 views

A question about the regularity of the Schrödinger equation

While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem, \begin{cases} -\Delta u+Vu=\lambda u, &\text{in } \Omega \\ \...
Du Xin's user avatar
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1 vote
1 answer
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Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data

I already asked the question on MSE, and have tried to figure it out myself. But the problem seems trickier than expected, so I guess MO is a better place to ask.. For the sake of completeness, I ...
Isaac's user avatar
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5 votes
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Elliptic regularity and Sobolev spaces

Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e. $$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$ where $a$ are ...
G. Blaickner's user avatar
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1 vote
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Continuity of a minimizing measure w.r.t a parameter

Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$. My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
BlueCharlie's user avatar
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Detailed estimate of the magnitude for the constant appearing in maximal regularity of the inhomogeneous heat equation

For any $T \in (0,\infty)$ and the $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ together with $1< p,q < \infty$, let us consider the following Cauchy problem: \begin{equation} \partial_t U - \alpha \...
Isaac's user avatar
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Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space

Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...
Javier Gargiulo's user avatar
1 vote
0 answers
106 views

Regularity of minimizing harmonic maps with no topological obstructions

So during (not really) my research I stumbled upon the following question, for which I could not find results in literature in any direction. It is not stated super precisely mathematically speaking, ...
Michele Caselli's user avatar
1 vote
0 answers
135 views

Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?

I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation. However, detailed proofs can ...
Isaac's user avatar
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1 vote
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A question about semigroups in a Heisenberg group

I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
Ilovemath's user avatar
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Brezis-Kato theorem

Let $n\geq 3$, and $u$ satisfies $$ -\Delta u=K(x)u^{\frac{n+2}{n-2}} \quad x\in B_1\setminus \{0\}, $$ where $|K(x)|\leq A$ in $B_1\setminus \{0\}$, and $u\geq 0$ in $B_1\setminus \{0\}$. Can we ...
Davidi Cone's user avatar
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91 views

Are Sobolev isometries in Minkowski space smooth

Let $\Omega\subset\mathbb{R}^d$ be an open regular domain and let $f\in W^{1,\infty}(\Omega;\mathbb{R}^d)$ satisfy that $df\in\operatorname{SO}(d)$ almost-everywhere. It was proved by Reshetnyak (in a ...
Raz Kupferman's user avatar
2 votes
0 answers
151 views

Regularity of a weak solution to an elliptic PDE with mixed boundary condition

I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition. Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ ...
sharpe's user avatar
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3 votes
0 answers
118 views

If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?

The question is as in the title. Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
Isaac's user avatar
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1 vote
0 answers
88 views

Non-existence of classical solutions of Hardy PDE

On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation $$ \begin{cases} u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
Ilovemath's user avatar
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1 vote
0 answers
199 views

Regularity up to the boundary of solutions of the heat equation

Given the heat problem: $$\begin{cases} \frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\ u(x,0)=u_0(x) & \forall x\in\Omega \\ u(x,t)=0 & \forall x\in\partial\...
joaquindt's user avatar
2 votes
1 answer
153 views

Global Hölder regularity

I am reading the book "Regularity theory for elliptic PDE" by Xavier Fernández-Real and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $...
Sean's user avatar
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0 answers
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Linear elliptic problems: Are gradient estimates preserved after perturbation?

(This question is a duplicate from here) We start with the linear elliptic PDE $$ -\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\ u=0 \quad\text{on}\ \partial\Omega $$ We assume that $\...
Muschkopp's user avatar
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3 votes
1 answer
265 views

Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?

Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that: $$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...
Bogdan's user avatar
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0 votes
1 answer
95 views

Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form

Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to $$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...
Simmetrico's user avatar
2 votes
1 answer
154 views

Optimal assumption on H^2 regularity

In many text book (Evans, Gilbarg-Trudinger for example) there is a classical result of interior regularity for weak solutions to a elliptic divergence problem $\rm{div}(A(x)u)=f$ in $\Omega\subset\...
Simmetrico's user avatar
4 votes
1 answer
146 views

Interior Sobolev regularity of parabolic solutions

In Evans book (and many others) there are a classic result about interior regularity in Sobolev spaces for solutions to uniformly elliptic problem (Theorem 1, p. 309). That is, let $\Omega\subset\...
Simmetrico's user avatar
1 vote
0 answers
31 views

Regularly of Neuman and Dirichlet problem

Let $\Omega \subset \mathbb{R}^n$, $n=2,3$ be a bounded domain with $\Gamma$ Lipschitz boundary (regularly) such that $\Gamma=\Gamma_D\cup\Gamma_N$. Assume that $h \in L^2(\Gamma)$ and $a \in C(\bar{\...
O Yassine's user avatar
5 votes
1 answer
335 views

Weak Hessian of the distance function

If $\Omega\subset\mathbf{R}^d$ has a smooth boundary it is known that the distance function $\mathrm{d}_\Omega:x\mapsto \mathrm{d}(x,\partial\Omega)$ is smooth on a neighborhood of $\partial\Omega$. ...
Ayman Moussa's user avatar
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4 votes
1 answer
426 views

Nonsmooth version of Hopf boundary point lemma

Let $$ Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u $$ be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite. Here I'm only considering smooth coefficients, and the domain $\...
leo monsaingeon's user avatar
3 votes
0 answers
51 views

Regularity of subelliptic eigenfunction on characteristic domain

Background: Consider the Hörmander vector fields $X=(X_1,\cdots,X_m)$ on $\mathbb{R}^n$, and the associated Dirichlet eigenvalue problem $$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
Houa's user avatar
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1 vote
0 answers
21 views

Regularity of solutions of a 2nd order singular integro-differential operator

I have trouble finding the regularity of the solutions to a particular equation. I define $$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
BlueCharlie's user avatar
2 votes
0 answers
164 views

Visualization of an oscillation lemma

How can one visualize Theorem 4.2 on page 31 of this paper by Seregin, Silvestre, Šverák and Zlatoš? On the other hand, I have a clear visualization of a related result about how oscillation decay ...
rick23's user avatar
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2 votes
0 answers
84 views

Maximal function to high power

Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ : $$ M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
Dorian's user avatar
  • 331
4 votes
1 answer
160 views

Smoothness of critical elliptic problem

I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H_0^1(\Omega)$ solution of $$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth ...
Math604's user avatar
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1 vote
0 answers
39 views

Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems. Is ...
valcofadden's user avatar
3 votes
0 answers
102 views

On the derivatives of the solutions of the heat equations with Neumann boundary condition

Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
sharpe's user avatar
  • 701
2 votes
1 answer
68 views

Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate. I am reading a paper of Brezis and Oswald about existence ...
Danilo Gregorin Afonso's user avatar
3 votes
0 answers
68 views

Does the incompressible Navier-Stokes equation have a smooth solution if the initial vorticity is smooth and $p$ integrable for any $p$?

Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$. If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $...
user99432's user avatar
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2 votes
0 answers
128 views

Smoothness of distance function induced by Finsler metric

Consider $\mathbb R^N$ endowed with a smooth Finsler metric $\phi:\mathbb R^N\times S^N\to (0,+\infty]$. The smoothness assumption are both on $\phi(x,\cdot)$ (being at least $C^{2,1}$) and $\phi(\...
DDG's user avatar
  • 21
3 votes
0 answers
83 views

Differentiability of a weak solution

Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...
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