Questions tagged [regularity]
regularity of solutions of PDEs.
266 questions
0 votes
0 answers
20 views
Hölder continuity up to the spatial boundary for singular and degenerate parabolic problems
I am interested in Hölder regularity of bounded solutions $u$ to the (one-dimensional) $p$-Laplace equation $$u_t = (|u_x|^{p-2} u_x)_x + b(x, t, u, u_x)$$ with $p \in (1, \infty)$ on a bounded ...
2 votes
2 answers
229 views
Necessary and sufficient conditions for coefficients of elliptic operator to obtain interior regularity
In the comments section in this other MSE question concerning a certain calculation left to the reader in Evans's Partial Differential Equations, @peek-a-boo and I were discussing the requirements on ...
0 votes
0 answers
49 views
On regularity for Hörmander's proof of classical well-posedness of systems of conservation laws in one spatial dimension
In Lectures on Nonlinear Hyperbolic Differential Equations by Lars Hörmander, a short proof is given of local-wellposedness of systems of classical well-posedness for quasilinear systems of ...
2 votes
0 answers
89 views
Elliptic equations in time-varying domains (non-cylindrical domains)
Suppose we are solving Poisson’s equation on a time-dependent spatial domain, i.e., we consider the steady-state heat equation at each time step while the domain evolves in time. That is, for each ...
1 vote
0 answers
47 views
Regularization property after transport
I have the following question: let's suppose we have some continous vector field $X$ on a compact surface and let's suppose that it is locally Lipschitz. Moreover, let's suppose that its flow $\Phi^...
4 votes
1 answer
167 views
Regularization of vector fields (convolution, semigroups...)
Let $M, N$ be smooth Riemannian manifolds. We consider a map $A : M \times N \rightarrow TM$ such that for all $z \in N, A(\cdot, z)$ is a vector field on $M$. Let us suppose that $A$ is smooth in the ...
2 votes
1 answer
131 views
Is there a simple criterion to determine when all weak solutions of an elliptic system are smooth?
I'm reading the paper by Mikhail Karpukhin and Daniel Stern, "Existence of harmonic maps and eigenvalue optimization in higher dimensions", Invent. Math. 236, No. 2, 713-778 (2024), ...
0 votes
0 answers
66 views
Regularity of distributional solutions of elliptic PDEs
I'm trying to generalize some classical theorems from geometric analysis and I faced the following problem: Let $(M,g)$ be a compact riemannian manifold and for $f:M \to \mathbb{R}$ continuous ...
0 votes
0 answers
59 views
Regularity of partially degenerate parabolic PDE
Let $T\in (0,\infty)$ and consider the following parabolic PDE: for all $(t,x,y)\in [0,T]\times \mathbb R^d\times \mathbb R^d$, $$\partial_t V -\Delta_x V - \sum_{i=1}^d H(\partial_{x_i} V+\partial_{...
2 votes
1 answer
111 views
$\text{div}(\mathbf {A}\nabla u)=0$ with bounded $\mathbf A$: least Holder continuity of $u$?
Let $u\in H^1(B_1)$ be a weak solution to $$\text{div}(\mathbf {A}\nabla u)=0 \qquad\text{in }B_1$$ where $\lambda \mathbf I\le\mathbf A\le\Lambda\mathbf{I}$. What is the least Holder continuity of $u$...
1 vote
0 answers
74 views
Reference request: solvability of reaction-diffusion equations
Consider the steady-state of the reaction-diffusion equation with an inhomogeneous Neumann boundary condition: $$ \begin{cases} D\Delta u +R(u,x)=0 & \text{in } U, \\ \partial_{\nu}u=f & \...
2 votes
0 answers
100 views
Maximal Regularity of Parabolic Equations in Besov Spaces and Heat Semigroup Characterization
I have been studying maximal regularity of parabolic equations in Besov spaces, particularly in the works of Ogawa and Shimizu: [Ogawa, Shimizu] End-point maximal ( L^1 )-regularity for the Cauchy ...
3 votes
0 answers
131 views
Does $\operatorname{curl}{v}\in L^2$ with $v$ divergence-free and ${v}\in L^\infty$ imply $\nabla v\in L^2$?
Let $v$ be a vector field in $L^\infty(\mathbb R^3, \mathbb R^3)$ such that $\operatorname{div}v=0$ and $\operatorname{curl} v\in L^2(\mathbb R^3, \mathbb R^3)$. Assuming moreover that $v$ goes to ...
1 vote
1 answer
229 views
Does $\operatorname{curl}v$ in $L^2$ imply $\nabla v$ in $L^2$ for a divergence-free vector field?
Let $v$ be a vector field in $\mathscr S'(\mathbb R^3, \mathbb R^3)$ such that $\operatorname{div}v=0$ and $\operatorname{curl} v\in L^2(\mathbb R^3, \mathbb R^3)$. Does that imply that all the ...
2 votes
0 answers
92 views
Is there a simpler proof for the $H^2$ boundary regularity estimate of a simple elliptic problem?
Given $ f \in L^2(U)$ and $\partial U$ is $C^2$. If $u \in H^1_0(U)$ is the unique weak solution to a simple elliptic problem with Dirichlet boundary condition $$ \begin{cases} - \Delta u ...
1 vote
0 answers
87 views
'Invert' perturbed vorticity equation to forced Euler system
Given the vorticity form of the Euler equations in $2D$ with stream function $\psi$ \begin{align} \omega_t + \nabla^\perp \psi \cdot \nabla\omega &= 0 \\ \Delta \psi = \omega \end{align} we know ...
0 votes
0 answers
106 views
Nonlinear quadratic Schrödinger equation with variable coefficients
Consider the following quadratic Schrödinger equation in $Q_T = \Omega \times [0,T]$: $$\begin{cases} i \partial_t u + \kappa(x,t) \Delta u + \beta(x,t) u^2 = f(x,t)\\ u(x,0) = ...
1 vote
1 answer
316 views
$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains
Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form $$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$ are there any elliptic ...
9 votes
2 answers
748 views
Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
2 votes
0 answers
111 views
3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$
Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$. Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
2 votes
1 answer
314 views
Can solution of heat equation become constant in finite time
Consider parabolic equation $$ u_t = u_{xx}, $$ where $x \in (0, 1)$, $t \in (0, T)$ with nonhomogenious Dirichlet boundary conditions. $$ u(0, t) = \psi_0(t) \in C^0, $$ $$ u(1, t) = \psi_1(t) \in C^...
8 votes
0 answers
157 views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
2 votes
1 answer
120 views
How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$
Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$. Assume ...
6 votes
1 answer
343 views
The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients
Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by $$Lu = \partial_i(a^{ij}...
1 vote
0 answers
82 views
Regularity of homogeneous random field in $\mathbb R^2$ and absolute continuity. Has this been generalized to the $n$-dimensional case?
What I am talking about is a rather old mathematical paper, published in Russian, from 1957. The name of the paper is “О линейном экстраполировании дискретного однородного случайного пoля”. I cannot ...
8 votes
0 answers
300 views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here. Let $C^k(\mathbb{R}^n$) be the ...
1 vote
0 answers
114 views
Regularity of the weak solution on the cube
Let $Q:=[0,1]^d$ and $g\in L^2(\Omega)$. Consider the PDE : $$ \begin{cases} -\Delta f=g & \text{in $\Omega$} \\ f\equiv 0 & \mbox{on $\partial \Omega$.} \end{cases} $$ I ...
2 votes
1 answer
163 views
Extension of Pohozaev's identity to the whole domain $\mathbb R^m$
I tried asking this question on Math SE but received no answers, even with a bounty. So I am asking it here. TLDR, I am trying to extend Pohozaev's identity on bounded domains to the unbounded domain $...
2 votes
0 answers
139 views
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 $\mathbb{R}^2$ ...
1 vote
0 answers
132 views
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 ...
5 votes
0 answers
146 views
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 ...
1 vote
0 answers
84 views
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^...
0 votes
0 answers
92 views
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 ...
0 votes
1 answer
306 views
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 $...
2 votes
0 answers
137 views
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 ...
10 votes
0 answers
466 views
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 \...
1 vote
0 answers
168 views
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}\...
4 votes
1 answer
218 views
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\\ ...
0 votes
0 answers
99 views
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(...
0 votes
0 answers
425 views
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 ...
2 votes
1 answer
143 views
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-...
1 vote
0 answers
185 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 ...
2 votes
0 answers
268 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 \\ \...
1 vote
1 answer
227 views
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 ...
4 votes
0 answers
296 views
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 ...
1 vote
0 answers
149 views
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 ...
1 vote
0 answers
178 views
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 ...
1 vote
0 answers
207 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, ...
1 vote
0 answers
281 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 ...
1 vote
0 answers
89 views
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 ...