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125 views

Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$

Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system ...
BigbearZzz's user avatar
  • 1,245
2 votes
1 answer
171 views

Mean value formula for fractional heat equation

For the solution $u(z) = u(t,x)$ of the heat equation $u_t -\Delta u = 0$ we have $$u(z_0) = \int_{\Omega_r(z_0)}u(z) K_r(z_0-z) dz,$$ where $$\Omega_r(z_0) = \left\{z \in \mathbb{R}^{N+1}: \Gamma(z_0-...
Zac's user avatar
  • 161
0 votes
1 answer
98 views

Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
Kernel's user avatar
  • 446
2 votes
0 answers
654 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
user270619's user avatar
1 vote
2 answers
106 views

Green function of symmetric stable process in dimension 1 and 2

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
user173196's user avatar
2 votes
1 answer
287 views

Reference request for semilinear PDEs in dimension 2

I am interested in the study of the (semi-linear, I suppose) equation $$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\ u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$ ...
UserA's user avatar
  • 597
1 vote
0 answers
105 views

Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?

Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
Jun's user avatar
  • 303
1 vote
1 answer
122 views

Existence and uniqueness for the equation $u_t + \nabla |u| = 0$

How does one prove the existence, uniqueness, and regularity for the equation $$u_t + \nabla_x |u| = 0 $$ with initial data $u(0,x) = u_0(x)$ and where the unknown function is $u:\mathbb (0,\infty)\...
rick23's user avatar
  • 41
1 vote
0 answers
36 views

Existence and uniqueness for fractional parabolic equation with transport term

Let us consider the problem \begin{equation} \begin{cases} u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\ u(x,0)=u_0(x) & \text{in } \...
user173196's user avatar
0 votes
0 answers
104 views

Rigorous energy estimate for advection-diffusion equation

Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and $q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$ $q \in [2,4], p \in [2,\infty] \text{ if } N = 1$ and consider the ...
user175203's user avatar
5 votes
1 answer
453 views

Seeking for references on some PDEs

This is not a technical mathematical question. I came across some PDEs with no references nor their names. $$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$ The above ...
Guy Fsone's user avatar
  • 1,101
1 vote
1 answer
195 views

Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$

Let us consider the problem $$ (-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$ where $s \in (0,1)$, $(-\Delta)^s$ is the fractional Laplace operator and $v:\mathbb R^n \to \...
user173196's user avatar
3 votes
1 answer
1k views

Friedrichs mollifiers and Sobolev spaces

$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
Carlos Esparza's user avatar
1 vote
0 answers
42 views

On the boundary integral of Neumann eigenfunctions

Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$...
student's user avatar
  • 1,350
5 votes
1 answer
297 views

A scaled fractional ''Sobolev inequality''

Does a fractional interpolation inequality similar to $$ \int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
Jay's user avatar
  • 109
7 votes
0 answers
351 views

Fractional Laplacian and chain rule

For the classical Laplacian, we have $$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$ for smooth functions $h$ and $u$. Does a similar chain rule hold (up to a reminder term) also for the ...
Zac's user avatar
  • 161
3 votes
1 answer
190 views

Laplace eigenfunction on a polygonal domain symmetric about an axis

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
user170399's user avatar
1 vote
0 answers
42 views

Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?

Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$ with Dirichlet boundary conditions. ...
Zac's user avatar
  • 161
1 vote
0 answers
66 views

Well-posedness of hyperbolic system with constant coefficients in finite domains

I'm studying the PDE $$ \frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0 $$ with $A_x, A_y, A_z$ being ...
viviaxenov's user avatar
3 votes
0 answers
127 views

Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?

My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider, $\dot{x}=Ax$, where $x$ is the infinite dimensional ...
Piyush Grover's user avatar
3 votes
0 answers
89 views

Error rate implying regularity

My question is a bit general/vague. It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...
user69642's user avatar
  • 778
3 votes
1 answer
148 views

Smoothing-Strichartz estimates for the heat-Schrodinger evolution

Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the ...
Capublanca's user avatar
2 votes
1 answer
178 views

References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition. In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
sharpe's user avatar
  • 721
3 votes
1 answer
299 views

Regularity and normal trace of "Hdiv" measures

In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$. I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
leo monsaingeon's user avatar
4 votes
0 answers
126 views

Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of Regular Lagrangian flow Filippov solution Caratheodory solution of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
Riku's user avatar
  • 839
1 vote
2 answers
424 views

Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$

Consider the problem $$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\ X(0,x) = x, &x \in \mathbb R \end{cases} $$ where $\chi$ denotes the ...
Riku's user avatar
  • 839
4 votes
1 answer
727 views

Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
user144878's user avatar
2 votes
0 answers
62 views

Existence and uniqueness for semilinear problem

Consider the following problem: $$-\Delta u + [(u)^+]^\alpha = 0,$$ where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
user avatar
4 votes
0 answers
93 views

Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian

What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^...
Glinka's user avatar
  • 381
1 vote
0 answers
52 views

Asymptotically periodic potentials

Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials?
Pádua's user avatar
  • 69
5 votes
1 answer
743 views

Eigenvalues and Domain of the Laplace-Beltrami Operator

Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
MartinG's user avatar
  • 51
1 vote
0 answers
62 views

Wellposedness of semilinear wave equation with discontinuous source

Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e. $$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $f$ is ...
Kei's user avatar
  • 277
2 votes
0 answers
95 views

Exp-decay estimate of Schrodinger equation

Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...
DLIN's user avatar
  • 1,915
1 vote
0 answers
75 views

Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
Kei's user avatar
  • 277
1 vote
1 answer
219 views

Harmonic functions vanishing on the boundary and distance function asymptotics

Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...
user avatar
3 votes
0 answers
376 views

Existence and uniqueness for reaction-diffusion equations

I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$ \begin{align*} &\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\ & u(0)=u_0\in L_2 \end{align*} where the ...
Oleg's user avatar
  • 931
2 votes
1 answer
391 views

Entropy solution for linear transport equation

Consider the transport equations $$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$ Can we define a notion of entropy solutions for (1) ...
Riku's user avatar
  • 839
4 votes
1 answer
370 views

Equivalence of viscosity and weak solutions for the Poisson equation

Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$. How does one prove that weak solutions are viscosity solutions and vice versa for the problem $$ \begin{cases} -\Delta u = f(x) & \...
user avatar
4 votes
0 answers
111 views

A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$

Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with $$ \nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u $$ but couldn't find a good ...
BigbearZzz's user avatar
  • 1,245
2 votes
0 answers
62 views

Differences among various index theories in critical point theory

Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones? the ...
Riku's user avatar
  • 839
1 vote
1 answer
247 views

Elliptic interface problem without conditions on the interface

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$. For a model case, consider a ball split in a smaller ball and an anulus. Consider the following elliptic ...
user avatar
4 votes
1 answer
322 views

compact embedding for Sobolev spaces

The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$ Is it possible to determine the ...
GabS's user avatar
  • 407
5 votes
1 answer
395 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
leo monsaingeon's user avatar
2 votes
0 answers
331 views

Sobolev embeddings for vector-valued functions

I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space. In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
Christopher A. Wong's user avatar
5 votes
3 answers
1k views

Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$

Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$ with initial data $u(0,\cdot) = u_0$. Fix $\alpha >0$. How can I estimate (without using explicitly ...
Riku's user avatar
  • 839
4 votes
0 answers
254 views

Lower semi-continuity of integration

I've found many papers characterizing the weak lower semi-continuity of $$ \Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx, $$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
ABIM's user avatar
  • 5,405
2 votes
2 answers
148 views

Solution of hyperbolic equations with $V^*$ data

Let $V\subset H\subset V^*$ a Hilbert triple and consider a 2nd order evolution equation of the form $$u''(t)+Au(t) = f(t)\quad \text{ in }\ L^2(0,T;V^*),$$ where $f\in\ L^2(0,T;H)$. Can we let $f\in ...
shall.i.am's user avatar
1 vote
1 answer
98 views

Sign-changing solutions for initial-boundary value problem for $\partial_t u + \partial^4_x u = 0$

Can you point out a reference for the fact that solutions for the initial-boundary value problem associated to $$\partial_t u + \partial^4_x u = 0$$ with $u(0,\cdot) >0$ can change sign (that is, ...
user avatar
3 votes
1 answer
1k views

Gagliardo-Nirenberg inequality for bounded domain

For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that $$ \|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}. $$ This is also true if $u\in W^{1,2}_0(\Omega)$ for a ...
BigbearZzz's user avatar
  • 1,245
2 votes
0 answers
93 views

Reference request: $|\partial_t u - \Delta u|\in L^p(\Omega^T)$ implies Holder estimate

I am currently reading a paper regarding harmonic flow between Riemannian manifolds. Let $\Omega$ be a Riemannian domain of dimension $2$, $\Omega^T=\Omega\times[0,T]$. The equation is $$ \partial_t ...
BigbearZzz's user avatar
  • 1,245