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-2 votes
2 answers
325 views

$f\in (W^{1,p}(\Omega)\cap C(\Omega) \cap L^{\infty}(\Omega))\setminus C(\bar{\Omega})$, $f=0$ on $\partial \Omega$ imply $f\in W^{1,p}_{0}(\Omega)$?

Q1: Let $p\geq 1$, and let $f\in W^{1,p}(\Omega)\cap C(\Omega)$. Assume also $f\in L^{\infty}(\Omega)$ and $f=0$ on $\partial \Omega$. Is it true that $f\in W^{1,p}_{0}(\Omega)$ even if $f\notin C(\...
14 votes
4 answers
1k views

Is every continuous microlocal operator a pseudo-differential operator?

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave ...
1 vote
1 answer
488 views

Motivation behind the parabolic metric

I've been reading some papers about parabolic evolution problems between manifolds. We want to study the behaviour of maps from the domain $(0,T)\times \Omega$, where $\Omega\subset \Bbb R^m$, to some ...
0 votes
1 answer
719 views

Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
5 votes
1 answer
472 views

Embedding theorem for anisotropic Sobolev spaces

Let $d=d_1+d_2$, $s_1,s_2>0$, $p>1$ and $(x_1,x_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$, $(\xi_1,\xi_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$. Define $$ W^{s_1,s_2}_{p}:=\left\{f: ...
3 votes
1 answer
289 views

A priori estimate of an inhomogeneous p-Laplace equation with Dirichlet boundary condition

I'm currently working on this Dirichlet problem: \begin{cases} div(\sigma |\nabla u|^{p-2} \nabla u) = f &\quad {in }~ \Omega\\ u = g &\quad in~\partial\Omega \end{cases} with $\sigma \in L^...
1 vote
0 answers
60 views

Existence of solutions to $\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f$

I'm looking for existence results for the equation $$\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f \quad \text{on the domain $[a,b]$}$$ for $u:[a,b] \to \mathbb{R}$, with either zero Dirichlet or ...
4 votes
1 answer
700 views

Is $L^1(\Omega)$ continuous embedded in the dual of $H^m(\Omega)$ $(m>\frac{d}{2})$?

Let $\Omega$ be a bounded domain of $R^d$ with Lipschitz boundary. If $m>\frac{d}{2}$, such that $H^m(\Omega)$ is continuously embedded in $L^\infty(\Omega)$. Is $L^1(\Omega)$ continuously embedded ...
1 vote
2 answers
219 views

Ground state has always constant sign?

Question: What hypothesis on the potential $V$ are required such that the ground state $\phi_0$ has constant sign? Consider the Schrödinger operator in 1 dimension with potential $V$: $$\mathcal{H}=-...
8 votes
4 answers
812 views

Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...
0 votes
0 answers
266 views

Embedding for the Bourgain spaces $X^{s,b}$

Where can I find embedding results for the Bourgain spaces $X^{s,b}$ (for a definition see the bottom of page 2 here). In particular, I'd like to know if, for $s$ sufficiently large, it is contained ...
1 vote
0 answers
102 views

domain dependence of best constant in inequality

Take $N \ge 3$ and consider the inequality $$ \| \nabla u\|_{L^N} \le C(\Omega) \| \Delta u \|_{L^\frac{N}{2}} $$ for all $ u \in W^{2,\frac{N}{2}}(\Omega) \cap W^{1,N}_0(\Omega)$ where $ \Omega$ is ...
2 votes
0 answers
229 views

Chain rule for Newton-derivative

I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon ...
-1 votes
1 answer
136 views

An elementary question about integration by parts! [closed]

Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
1 vote
1 answer
728 views

Elliptic regularity of Laplace-Beltrami operator on a manifold

I am currently trying to prove an elliptic regularity type result for the Laplace Beltrami operator $\Delta_g$ on a Riemannian manifold $(M^n,g)$. As a matter of convention, I will assume $\Delta_g$ ...
11 votes
1 answer
668 views

Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
8 votes
2 answers
2k views

when a pseudo-differential operators to be compact?

In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $ My ...
2 votes
0 answers
78 views

Generalization of supersymmetry to dimension 3

in two dimensions there is a simple trick to study the spectrum of operators of the form $$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$ The trick is to ...
2 votes
1 answer
244 views

Smoothness of distributions defined by oscillation integrals

In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement. Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
3 votes
2 answers
1k views

Orthogonality to harmonic functions

Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define $a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$ $b_n=f\Delta^{...
0 votes
1 answer
104 views

Poisson Equation across a Hypersurface [closed]

Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem $ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
2 votes
0 answers
683 views

Laplace problem with Robin boundary condition on a wedge

I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let \begin{equation*} \Omega = ...
1 vote
3 answers
166 views

Let $u_n\in\mathcal{D}'(\mathbb{R}^n)$ have $u_n\to0$ where $u_n\in C_c^\infty$ have uniformly compact support. Does $u_n\to0$ in $C_c^\infty$? [closed]

Suppose we have functions $u_n\in C_c^\infty(\mathbb{R}^d)$ with support all lying in $B(0,R)$, and suppose $u_n\to 0 $ in $\mathcal{D}'(\mathbb{R}^n)$, i.e. for all $\eta\in C_c^\infty(\mathbb{R}^d)$,...
2 votes
1 answer
713 views

Reference request: numerical analysis of PDEs and integro partial differential equations

I'm very new to the field of numerical analysis of PDE and integro partial differential equations. My advisor (who is not a specialist in this area) highly recommended to read Randall J. LeVeque'...
0 votes
1 answer
218 views

Heat semigroup dissipative

Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition. On $L^2$ it would be completely trivial, but ...
1 vote
0 answers
198 views

Morrey space is Banach space

I'm working with Morrey spaces, which are the spaces $$L^{p,\lambda}(\Omega):= \left\{ u \in L^1_{loc}(\Omega): \sup_{x \in \Omega, r > 0} r^{-\lambda}\int_{B(x,r)\cap \Omega}|u(y)|^pdy< \infty\...
4 votes
1 answer
367 views

Dissipative operator on Banach spaces

An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$ $$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$ On a Hilbert space this is ...
2 votes
1 answer
1k views

Proof of Agmon's inequality in $\mathbb{R}^3$

According to Wikipedia, Agmon's inequality provides a bound on the $L^\infty$ norm of a $H^2$ function on a (regular) subset of $\mathbb{R}^3$. In the book of JC Robinson et al. "The Three-...
1 vote
1 answer
350 views

Functions orthogonal to harmonic functions

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose $\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...
1 vote
1 answer
151 views

Global wellposedness of the Cauchy problem for a third order PDE

Consider $$u_t-\alpha u_{xx} - \beta u_{xxx} = f(u_x)$$ with initial condition $u(0,x) = u_0(x)$, where $\alpha>0$, $\beta \in \mathbb{R}$, $u_0 \in C^\infty(\mathbb{R})$, and $f$ is Lipschitz (...
4 votes
0 answers
164 views

A modern reference for the "Intermediate Derivatives Theorem"

In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows: Intermediate Derivative Theorem: Let $X\subset ...
2 votes
2 answers
304 views

Why is the ellipticity condition important in the theory of viscosity solutions?

A fundamental assumption in User's guide (p. 2) is that the operator $F$ should be proper. However, the role of this monotonicity condition (and especially of the "ellipticity condition" part) is ...
2 votes
0 answers
142 views

Self-adjointness on Banach spaces

Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem. Now, if we have an unbounded ...
11 votes
2 answers
1k views

Concentration compactness. Can this concept be stated in a theorem?

I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk. When I approached the speaker ...
3 votes
1 answer
148 views

Prove existence of continuous function on $(0,1)$ with special properties [closed]

Consider the interval $I=(0,1)$ and let $f,g$ be two linearly independent continuous functions on $[0,1]$. I am asking if there is a continuous function $h$ such that $$\int_0^1 h(s) f(s) ds=0$$ $$...
1 vote
1 answer
335 views

Orthonormal basis and decay

Edit: I added smoothness, hoping to simplify the problem with this additional assumption. Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
1 vote
1 answer
139 views

Compactly supported functions and projections

Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f_n$ on $\Omega$ normalized to one (in the $L^2$ sense). Then, these functions span a ...
2 votes
0 answers
2k views

Reference for a proof of the Gagliardo-Nirenberg Interpolation Inequality?

In the book Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Cherrier and Milani, Theorem 1.5.2, we are given the following version of the GN interpolation inequality: Let $\Omega\...
5 votes
1 answer
379 views

References on the obstacle problem for the heat equation

Can you point out some references that deal with the obstacle problem for the heat equation? $$(OP) \quad\begin{cases} \max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \...
2 votes
2 answers
406 views

Solution to semilinear heat equation at $t=0$: $u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$

Consider the following Cauchy problem $$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$ Suppose that $u \...
12 votes
3 answers
2k views

Reference request: Simple facts about vector-valued Sobolev space

Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
4 votes
1 answer
2k views

Crandall & Rabinowitz Theorem, bifurcation curves

Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type: $$ F(\lambda,u) = \lambda u - G(u) = 0, $$ where $G \in C^1(X,X)...
1 vote
0 answers
63 views

Supnorm problem involving kernel of Cauchy problem

Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem $$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...
3 votes
0 answers
125 views

Partial regularity for transmission problem in corner domains

Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
2 votes
1 answer
219 views

Let $X\subset Y$ be a dense inclusion of reflexive Banach spaces. Then is $C_c^\infty(\Omega;X)\subset C_c^\infty(\Omega;Y)$ dense?

Let $X$, $Y$ be reflexive Banach spaces, and let $\imath:X\hookrightarrow Y$ be a bounded inclusion with dense image. Then for any domain $\Omega\subset\Bbb R^n$ we may define $C_{\mathrm c}^\infty(\...
4 votes
1 answer
562 views

Fundamental solutions for degenerate elliptic equations

I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elliptic equations $L = -divA\...
1 vote
0 answers
79 views

Time dependent Hamiltonians

I'm studying time dependent perturbation theory on Reed-Simon book "Method of modern mathematical physics, II". If one considers an Hamiltonian of the form $$H(t)=H_0+V(t)$$ the corresponding formal ...
5 votes
2 answers
980 views

Symbol of the Laplace-Beltrami on $\mathbb{S}^2$

This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e. A differential operator $P=\sum_{|\...
0 votes
1 answer
268 views

Linear operator has one-dimensional kernel

Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
10 votes
3 answers
1k views

References: spectral analysis of the Laplacian operator

I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature....

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