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Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
Cauchy's Sequence's user avatar
3 votes
1 answer
741 views

About radial Sobolev inequality (Strauss Lemma)

As shown in Strauss: Existence of solitary waves in higher dimensions, Strauss introduces the Stauss lemma. Precisely speaking, we have the following theorem: Theorem Let $N \ge 2$, every radial ...
Tao's user avatar
  • 429
3 votes
1 answer
164 views

Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
BigbearZzz's user avatar
  • 1,245
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
3 votes
1 answer
441 views

Courant nodal domain theorem for fractional Laplacian

Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$. That is, $\...
Voliar's user avatar
  • 205
3 votes
1 answer
302 views

Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$

Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be ...
Ribhu's user avatar
  • 407
3 votes
1 answer
328 views

Typical elements of the space $\mathring {L^k_p}(\Omega)$

In the book Sobolev Spaces with Application of Maz'ya, $\mathring {L^k_p}(\Omega)$ is defined to be the completion of $\mathcal D(\Bbb R^n)$ under the norm $||\nabla^ku||_{L_p(\Omega)}$. For nice ...
BigbearZzz's user avatar
  • 1,245
3 votes
1 answer
415 views

Continuity of convolution on L¹

Question Let $f,g\in L^1(\mathbb{R}^n)$ be nonnegative and assume $\int f\,g \; dx <\infty$. Let $\Phi_\varepsilon$ be the heat kernel at time $\varepsilon$. Denote with $f_\varepsilon = f \ast \...
André Schlichting's user avatar
3 votes
1 answer
762 views

Approximate Fréchet derivative of PDE solution operator

Let's assume that $F$ is the solution operator of some PDE problem, i.e. $F$ maps some parameter or boundary condition function to the solution of a given PDE. Let also be $\Pi$ be the evaluation ...
Philipp Wacker's user avatar
3 votes
1 answer
531 views

An argument in the proof of a compactness theorem

In the proof of a compactness theorem involving fractional derivatives in Temam's Navier-Stokes Equations, an argument as the following is made. Suppose $X_0,X,X_1$ are Hilbert spaces such that ...
user avatar
3 votes
1 answer
326 views

Spectral multipliers vis-a-vis Differential geometry

Let us mention two papers for examples: this one by Seeger and Sogge and this by Cheeger, Gromov and Taylor. One can also mention papers by Stein, for example, this one. There are also many others of ...
MBM's user avatar
  • 141
3 votes
1 answer
701 views

Doubling of variables method for parabolic equations

Does anyone have a reference that explains the technique of doubling of variables as introduced by Kruzkov? It seems to be a necessary tool for contraction estimates when we have weak solutions. ...
mathias_l's user avatar
  • 145
3 votes
1 answer
302 views

Techniques to show existence for a PDE with dynamic boundary condition

Let $\Omega$ be a bounded domain. I am looking for techniques to show existence of solutions to dynamic boundary problems of the form $$\Delta u = 0 \quad\text{on}\quad \Omega \times (0,T)\\ \qquad\...
TheBook's user avatar
  • 155
3 votes
1 answer
495 views

Inequality in the Sobolev space $H^1$

I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ^2\bigg)^{\frac{q}{2}-a}+\frac{c}{r^{2a}}\bigg(\int_{B_r}...
user avatar
3 votes
3 answers
228 views

References for well-posedness of weak solutions to Stefan problem

Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)? ...
soup's user avatar
  • 307
3 votes
1 answer
518 views

A closed extension of the Laplace operator with respect to the supremum norm

Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
user36746's user avatar
3 votes
1 answer
733 views

Trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
michael_faber's user avatar
3 votes
1 answer
393 views

A Sobolev-type inequality with weights

In the study of a particular PDE I found myself wanting to prove the following inequality: $( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r |f''|^2]...
Matt Cooper's user avatar
3 votes
2 answers
642 views

Localization of Laplacian eigenfunction on the unit square?

Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
Denis Grebenkov's user avatar
3 votes
1 answer
2k views

Existence of solution for Poisson problem with pure Neumann BCs

Hello all, Does the following boundary value problem admit unique solutions $q$: $- \Delta q + \beta q = f$, $x \in \Omega$ $ \nabla q \cdot \vec{n} = g $, $x \in \Gamma := \partial \Omega$, ...
Mihai's user avatar
  • 53
3 votes
1 answer
182 views

Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product

Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing). For any pair of ...
Isaac's user avatar
  • 3,477
3 votes
1 answer
370 views

Is Brascamp-Lieb inequality on the sphere applicable for these functions for some $1\leq p<2$

My question is on Brascamp-Lieb-inequality on the Euclidean sphere (which can be viewed as an analogue of Young's inequality on the sphere) obtained in [1]. (See also this question: Brascamp-Lieb ...
Medo's user avatar
  • 852
3 votes
1 answer
302 views

Convergence of a level set when $f^n\to f$ in $C^1$ sense

Let $f^n$ be a family of $C^1$ functions and $f(x)=-|x|^2+1$ such that $$f^n\to f$$ in $C^1$ sense as $\varepsilon\to 0$. I want to ask that does the level set $\{f^n=0\}$ converges to $\{f=0\}$ in ...
W.J.'s user avatar
  • 379
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
3 votes
1 answer
300 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
3 votes
1 answer
541 views

regularity of p-harmonic functions

We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$. Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
user143090's user avatar
3 votes
1 answer
255 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
user avatar
3 votes
1 answer
876 views

Is Quantum Mechanics (norm)-consistent?

I edited a few small comments to the question in order to make it perhaps more comprehensible. Today I came across the following question in quantum mechanics. In Quantum mechanics it is common to ...
Hilbertspace's user avatar
3 votes
1 answer
431 views

Can I approximate a function of bounded variation with orthogonal polynomial?

Let function $u\in BV(\Omega)$ be a function of bounded variation and $\Omega\subset \mathbb R^2$ be a smooth domain. I know it is possible to approximate function $u$ with polynomials, i.e., $$ u = \...
wingsofpanda's user avatar
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^{...
A random mathematician's user avatar
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 $$\int_0^1 h(s) f(s) ds=0$$ $$...
Zinkin's user avatar
  • 501
3 votes
1 answer
187 views

Free quantum evolution operator on Sobolev space

I am not a mathematician, but would like really like to get some confirmation on the things I am doing here. Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then ...
plain's user avatar
  • 95
3 votes
2 answers
397 views

The space $L^p(\partial\Omega)$ in cited references

The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
user avatar
3 votes
1 answer
212 views

A question on the Frechet derivative

Suppose the derivative of a functional is given by \begin{equation*} \int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in W_0^{1,p}(\...
Alexander's user avatar
  • 157
3 votes
1 answer
845 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
student's user avatar
  • 31
3 votes
1 answer
381 views

What are the applications of Grillakis Shatah and Strauss paper?

I am studying the following paper. Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. ...
Marcos Rojo's user avatar
3 votes
1 answer
138 views

$C_0$ semigroups on parameterized Banach spaces or moving domains

Is there any literature corresponding to one or two-parameter semigroups such that e.g. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$? I ...
assa888's user avatar
  • 153
3 votes
1 answer
1k views

Strong maximum principle for weak solutions

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...
LapLace's user avatar
  • 173
3 votes
1 answer
207 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
Giorgio's user avatar
  • 31
3 votes
1 answer
215 views

Checking initial data in parabolic PDE with no control on time derivative

It is possible to define a weak solution of a parabolic PDE $$u_t - Au = f$$ $$u(0) = u_0$$ as $u \in L^2(0,T;H^1)$ such that $$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega Au(t)\varphi(...
TheBook's user avatar
  • 155
3 votes
1 answer
344 views

Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$ Further assume a solvability condition $$\int_\Omega f ~\mathrm{d}\...
Elias Ka's user avatar
3 votes
1 answer
251 views

Null sets in PDE

Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = \...
maximumtag's user avatar
3 votes
1 answer
1k views

Showing a singular integral operator takes Hölder continuous functions to Hölder continuous functions of the same order

I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...
marcpal's user avatar
  • 31
3 votes
1 answer
354 views

Solvability for constant-coefficient partial differential operators

Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...
Kevin McLeod's user avatar
3 votes
1 answer
747 views

Reference request: Anisotropic Sobolev spaces

Hello, I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying $ \| f \|_p < \infty, \|Df \|_q < \infty, $ where $p \ne q$ (as ...
Idempotent's user avatar
3 votes
1 answer
949 views

ODE continuous dependence on parameters to PDE

I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
Bloop's user avatar
  • 55
3 votes
1 answer
261 views

The continuous dependence of the Green's function on a domain

Let $\Omega\in\mathbb{R}^2$ be a smooth bounded domain and $G(x,y)$ be the Green's function of $-\Delta$ in $\Omega$ with zero Dirichlet condition. Clearly $G(x,y)=-\frac{1}{2\pi}\ln|x-y|-h(x,y)$, ...
W.J.'s user avatar
  • 379
3 votes
2 answers
129 views

Link between controllability of ODEs and controllability of transport equations

What is the relationship between the controllability of the ODE $$\dot x(t) = v(x) + u(t)$$ using a control $u$ and the controllabilty of the transport equation $$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
Riku's user avatar
  • 839
3 votes
1 answer
552 views

Lax-Milgram and the existence of solution to parabolic equation

I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not ...
mnmn1993's user avatar
3 votes
1 answer
107 views

Finiteness of Schatten $p$-norm of truncated free resolvent

Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$. For $p \geq 1$, let ...
user271621's user avatar

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