For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

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176 views

The $p$-Laplacian is strongly monotone

I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$ E[u]=\frac{1}{p}\int_\Omega \lvert\nabla ...
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0answers
123 views

The equivalent definition of $W_0^{1,\infty}(\Omega)$

Usually, for $1\leq p<\infty$, we define $W_0^{1,p}(\Omega)$, where $\Omega$ is open bounded smooth boundary, by taking the closure of $C_c^\infty(\Omega)$ under $W^{1,p}$ norm. However, we don't ...
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2answers
120 views

Counter example for Poincare inequality does not hold on unbounded domain

The Poincare inequality states that if domain $\Omega$ is bounded in one direction by length $d>0$ then for any $u\in W_0^{1,p}(\Omega)$ we have $$ \int_\Omega|u|^p\,dx\leq ...
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1answer
95 views

What is the dense subset in $H_0^1(\Omega)\cap H^2(\Omega)$

I came across this problem when I try to prove that for space $H(\Omega):=H_0^1(\Omega)\cap H^2(\Omega)$, where $\Omega$ is open bounded with nice boundary, then the norm $\|u\|_1:=\|\Delta u\|_{L^2}$ ...
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1answer
62 views

An linear elliptic PDE, why it has these properties?

See this image (from this work) The existence is done through Lax-Milgram (at least for $\sigma = \frac{1}{2}$), I think. However, why the author only includes the gradient in defining $H^1$? Is it ...
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1answer
39 views

uniform constant and radius bound for poincare type inequalities On compact manifolds

I have a clarification question. If we have a Riemannian compact manifold $M$, then there exists constants c and $r_{0}$ such that for any radius r < $r_{0}$ we have $$ \bigg(\frac{1}{|B_{r}(x)|} ...
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1answer
131 views

Are all functions in the Sobolev space $W_0^{1,2}(\Omega)$ continuous and bounded?

Are all function in $W_0^{1,2}(\Omega)$, $\Omega$ being a bounded domain in $\mathbb{R}^n$, $n \geq 2$, continuous and bounded w.r.t. $|.|$?. In other words, given $u\in W_0^{1,2}$ can one say that ...
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80 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
5
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1answer
65 views

Geometric Interpretation of Weak Derivative

As we know, classic derivative $f'(x)$ of a function $f(x)$ can be interpreted as the rate of change of function $f$ in each point $x.$ How about weak derivative? Since it is defined through integral ...
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2answers
116 views

Dirichlet Principle in Sobolev Spaces

According to Zeidler, 1995, in his book "Applied Functional Analysis: Application to Mathematical Physics". Dirichlet problem is a problem to minimize $$F(u)=\frac{1}{2}\int_G(u')^2\ dx-\int_G fu\ ...
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0answers
217 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial ...
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1answer
54 views

Estimation the $L^p$ norm of $u$ by using trace and gradient.

Given $\Omega\subset \mathbb R^N$ open bounded with nice boundary. Then for $u\in W^{1,p}(\Omega)$, $1\leq p\leq \infty$, we have $$\|u\|_{L^p(\Omega)}\leq C(\|T[u]\|_{L^p(\partial\Omega)}+\|\nabla ...
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1answer
52 views

Does convolution of two functions in $H^s(\mathbb{R})$ belong to $H^{2s}(\mathbb{R})$?

Let $f$, $g$ be two density functions and assume that $f,g\in H^s(\mathbb{R})$, $s>\frac{1}{2}$, where $${H^s}(\mathbb{R}) = \left\{ {u:\int_{ - \infty }^\infty {{{\left| {\hat u\left( t \right)} ...
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1answer
71 views

The essential sup over the boundary VS the trace of a Sobolev function

Given $\Omega\subset \mathbb R^N$ is open bounded, nice boundary, and $u\in H^1(\Omega)$. We say that $u\leq \alpha$ on $\partial \Omega$ for a constant $\alpha$ if $(u-\alpha)^+\in H_0^1(\Omega)$, ...
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1answer
110 views

Application of Fubini's theorem (in a proof of energy minimizing harmonic maps)

Let $u\in H^1(B_1,S^k)$, where $B_1$ is the open unit ball in $\mathbb{R}^n$ and $S^k$ is the unit sphere in $\mathbb{R}^{k+1}$. Suppose that $u$ is a minimizer for the Dirichlet energy functional $$ ...
2
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1answer
71 views

Is there an unbounded integrable function with integrable derivative in $(0,1)$?

I wonder if there is a differentiable unbounded function $f\in L^1(0,1)$ with $f'\in L^1(0,1)$. The elementary examples $x^\alpha$ or $\log x$ suggest that my question should be answered negatively. ...
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1answer
92 views

Intuitive question about the trace operator (Sobolev spaces)

Let $\Omega$ an open and bounded domain in $R^n$ . Let $u \in W^{1,p} (\Omega) \cap L^{\infty}(\Omega)$ $(2 \leq p < \infty)$ . Let $B \subset \subset \Omega$ a open ball and consider $u_1$ the ...
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2answers
75 views

A modified version of Poincare inequality

We know the general version of Poincare inequality: $$ \int_\Omega |u-u_\Omega|^2dx\leq C\int_\Omega|\nabla u|^2dx,\quad \forall u\in W^{1,2}(\Omega), $$ where $u_\Omega$ is the average of $u$ over ...
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1answer
67 views

maximum principle for $p$-Laplace equation

Consider $\Omega \subset R^n$ a bounded domain. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$. Let $u \in W^{1,p}(\Omega)$ with $\Delta_p u = 0$ in $\Omega$ with $u - \varphi\in ...
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132 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
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1answer
65 views

Showing that $a(\cdot,\cdot)$ is coercive

I am working on a problem and I have the weak formulation of Poisson's problem in $2$ spatial dimensions i.e. $u = u(x,y)$: $$a(u,v) = \ell(v) $$ where $$a(u,v)=\int_{\Omega}\nabla u\nabla ...
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0answers
51 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
2
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1answer
78 views

Is $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$ dense in $H^1(\Omega)$?

Can it be true that the space $$\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$$ is dense in $H^1(\Omega)$? If so, please give me a reference to this. Every $u \in H^1$ has $\Delta u \in ...
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0answers
41 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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0answers
99 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
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1answer
61 views

Trace map from $H^1$ into $H^{\frac 12}$, does this statement imply another?

Consider trace map $T:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ on a sufficiently smooth domain $\Omega$. It has a partial inverse $E$. If we have the statement $$F(u,Eu) = 0\quad\text{for all ...
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1answer
53 views

Help with proof that $H^{1}(\mathbb{R})$ is closed under multiplication.

Edit: Prove that if $u,v \in H^{1}(\mathbb{R})$ then $uv \in H^{1}(\mathbb{R})$. My idea is to approximate with functions in $C^{\infty}(\mathbb{R})$ with compact support. Let $u,v \in ...
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1answer
37 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
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1answer
95 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
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0answers
36 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
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29 views

Second order equation and regularity

Let $U$ be an bounded open set in $\mathbb R^3$ and ${V}\subset \subset{W} \subset \subset {U}$. Let $u \in {H^1}(U)$, and $f\in L^{2}(U)$ satisfies $$\int_{\mathbb R^3} \nabla u(x). \nabla \varphi ...
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0answers
59 views

Integration by parts in Sobolev space

I'm looking for a reference of the following fact (if it is true...): if $u\in W^{1,1}(\Omega)$ and $v \in W^{1,\infty}(\Omega)$ ($\Omega$ a open subset of $\mathbb{R}^n$ ($n \ge 1$) with a regular ...
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1answer
61 views

calculate weak derivate of $|x-2|^2$

Let $u$ be a function with $u(x):=|x-2|^2$ on $I:=(-1,1)$. I want to test whether $u \in H^2(I) \backslash H^3(I)$. Let $\phi$ be in $C_0^\infty(I)$. Then: $T_u(\phi '') = \int_{-1}^1 |x-2|^2 ...
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43 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p ...
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0answers
58 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
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1answer
41 views

Basic question about weak solution of p-Laplace equation

Let $p \geq 2$ and $B(x_0, d)$ an open ball in $R^n$ ($n \geq 2$). Let $u \in W^{1,p}(B(x_0,d))$ a weak solution of $\Delta_p u = f$ , $u = 0 , \ on \ \partial B(x_0,d)$, that is, $\int_{B(x_0,d)} ...
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1answer
62 views

Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I ...
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1answer
99 views

Is fractional order Sobolev spaces reflexive?

Let $0<s<1$, we define $$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$ with norm $$ \|u\|:=\left(\int_{\Omega} ...
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1answer
91 views

The Coercivity of uniformly positive definite Matrix of Sobolev function

For $u=(u^1,\ldots, u^N)\in W^{1,2}(\Omega,R^N)$ where $\Omega$ is bounded. We define $$ E[u]=\int_\Omega g_{ij}(u)\nabla u^i\nabla u^jdx$$ where $G=(g_{ij})_{1\leq i,j\leq N}$ is an given uniformly ...
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1answer
43 views

About a Morrey's type inequality

Let $\Omega \subset R^n$ an open bounded domain and consider $B_r(x_0) \subset \Omega$ an open ball. Let $u \in W^{1,p}(\Omega)$ ($p \geq 2$). Let $s > n$ and suppose that $\int_{B_r(x_0)} |\nabla ...
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1answer
58 views

A property of sobolev spaces

Let $W^{k,p}(\Omega):=\{y\in L^p(\Omega) : D^{\alpha}y\in L^p(\Omega)$ for all $|\alpha|\leq k\}$ I want to prove now that: (1) $u \in W^{1,2}(\mathbb R)$ is equivalent to (2) $u \in L^2(\mathbb ...
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66 views

Sobolev estimation of second derivative against Laplacian and higher terms

Given $u \in H^2(\Omega)$ (and $\Omega \subseteq \mathbb{R}^n$ with appropriate properties) is there a way to estimate the norm of the second derivative $\Vert D^2 u\Vert_{L^2(\Omega)}^2$ against the ...
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0answers
27 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
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0answers
57 views

counterexample to strauss inequality

I am looking for a counterexample to Strauss inequality in dimension 1, where it supposedly fails. How can one construct an $H^1(\mathbb{R})$ function which does not decay at infinity, for which for ...
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1answer
81 views

Existence of weak derivative

Can a uniformly continuous function have a weak derivative?. In other words can $C_{unif.~cont.}$ be continuously be embedded in $W^{1,2}(\Omega)$.?
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1answer
29 views

Is Hlawkas Inequality holds for sobolev space

im wondring is that inequality holds for any functionnal space such as sobolev space and if it's true how we can write it in that space /HlawkasInequality any help would be apperciated
3
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1answer
417 views

The Courant Min-Max theorem of elliptic pdes.

This is an exercise function Evans PDE book, Chapter 6. The theorem states that for $Lu:=-\text{div}(A\cdot\nabla u)+cu$ where $c\geq 0$, we have the eigenvalue of $L$ can be written in the following ...
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0answers
41 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
3
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0answers
44 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
1
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1answer
59 views

Is this derivative somehow bounded?

I have a function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is a test function and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by $f(x) = \phi(\frac{\|x\|}{n})$. Now if I take any ...