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1 vote
0 answers
84 views

Does sets of positive capacity rule out constant functions?

Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...
2 votes
0 answers
54 views

Distance between a Hölder function and a Sobolev ball

Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively. My question ...
5 votes
2 answers
364 views

Euler-Lagrange equations for minimizer of energy with indicator function

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
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)? ...
1 vote
0 answers
78 views

Trace theorem for $L^2([0,1]; H^k(S^2))$

Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer. Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
8 votes
3 answers
1k views

Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem. Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
2 votes
2 answers
235 views

$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has $$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
2 votes
2 answers
197 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
3 votes
1 answer
296 views

Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type $$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ ...
6 votes
1 answer
382 views

Sobolev embedding theorems on manifolds

I had asked the following question on math.stackexchange but did not get any response: I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
7 votes
1 answer
652 views

Extending Hölder functions

I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
2 votes
0 answers
124 views

Uniqueness in interpolation of Hilbert spaces

I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
2 votes
0 answers
279 views

Relationship between $p$-capacity and Riesz $s$-capacity of a set

What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
2 votes
2 answers
451 views

The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$

Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
4 votes
2 answers
391 views

Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$ Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$. Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then $$ u(x)...
3 votes
1 answer
466 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
1 vote
1 answer
404 views

Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? [closed]

Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
4 votes
0 answers
68 views

Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
2 votes
1 answer
351 views

References on duality of fractional order Sobolev spaces

I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
2 votes
0 answers
149 views

Reference for weighted Sobolev spaces

I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
1 vote
0 answers
110 views

Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space

$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set. For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
1 vote
0 answers
79 views

Reference for smoothness of Nemytskii operator on fractional Sobolev spaces

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator $$ \big(N_\varphi x\big)(t)=\varphi\big(x(t)\big) $$ for $x\in H^s(T^d)$, the ...
2 votes
0 answers
559 views

Multiplication in Sobolev space with negative exponent

My initial problem is the following: I would like to estimate $\lVert f^2\rVert_{H^{-2}}$ in the sense of Sobolev embeddings, where $f:\Omega \rightarrow \mathbb{R}$ is a function defined on a bounded ...
1 vote
1 answer
277 views

Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$

Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets $$\begin{align*} S_1 &= \left\{ \begin{...
1 vote
0 answers
56 views

Smooth approximation in Sobolev spaces for surfaces with boundary

Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
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 ...
3 votes
0 answers
181 views

Variational problems living in two different Sobolev spaces

Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type: $$\inf_{u,v}\int_{\Omega} ...
1 vote
1 answer
116 views

uniform convergence of $H^r$ projectors on compact sets?

Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
2 votes
0 answers
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 ...
0 votes
1 answer
96 views

Interpolated Sobolev norm inequality

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
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 ...
2 votes
0 answers
72 views

Product of Besov and Lorentz functions

Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound $$ \|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
4 votes
0 answers
176 views

If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$

Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
4 votes
1 answer
256 views

Regularity of Nemitskii maps on Sobolev spaces

Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$). Let $X=W^{1,p}(\Omega)$ with $p>1$ be the ...
9 votes
1 answer
1k views

Traces of Sobolev spaces

Is there a simple proof of the following fact? Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset W^{1-\frac{1}{n},n}(\...
6 votes
1 answer
320 views

Stability of fractional Sobolev spaces under diffeomorphisms

Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
2 votes
0 answers
207 views

Smoothing properties of analytic semigroups

Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates $\|T_p(t)f\|_{...
5 votes
2 answers
1k views

Rates of convergence of mollifiers with Sobolev norms on manifold

Let $M$ be a smooth compact Riemannian manifold of dimension $n$, and let $H^s_p(M)$ for $s\in \mathbb{R}$ be the fractional Sobolev space of order $s$ on the manifold (defined for instance through ...
3 votes
1 answer
695 views

Continuous embedding between parabolic Sobolev spaces

I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct? Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
4 votes
0 answers
102 views

$W^{k,p}$ and Holder regularity for linear elliptic systems with Neumann boundary data

I'm looking for a text or paper that discusses regularity in the Sobolev and Holder sense for general linear elliptic systems of PDEs on bounded domains with Neumann boundary data. The book by ...
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}...
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 ...
1 vote
0 answers
198 views

Sobolev embedding in complete manifold

Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose that $(M,g)$ admits a bounded geometry. Q Can we show that for $k-\frac{m}{p}\geq l-\frac{m}{q}$, we ...
5 votes
3 answers
1k views

Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous? I define $H^{\frac 1 2}(\...
2 votes
0 answers
255 views

Sobolev Multiplication on non-compact manifold

We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
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 ...
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 ...
8 votes
0 answers
260 views

Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$

I have found the following claim made very clearly at least once in the published literature (see below): Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
5 votes
0 answers
445 views

Why are functions with vanishing normal derivative dense in smooth functions?

Question Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm? Here I define $...
3 votes
1 answer
579 views

What's the predual of Holder continuous function spaces?

Given Holder space $C^{\alpha}(\mathbb{R}^n)$, does there exist a Banach space $X$ such that the dual of $X$ is $C^{\alpha}(\mathbb{R}^n)$? What I can imagine is that such $X$ must contain the ...