0
votes
0answers
9 views

reference request about sobolev space and BV space

I am studying Sobolev Space and BV space by using Leoni's and Evans & Gariepy's book. I was wondering that where can I find some explicit example and some computational question of those space. A ...
5
votes
1answer
129 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
2
votes
0answers
37 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
2
votes
1answer
42 views

Global bounded solution of $u_{tt}=\Delta u-mu+h$ in the Hilbert space $X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Let $\Omega$ be an open subset of $\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ...
5
votes
0answers
45 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
3
votes
1answer
71 views

Characterization of weak solution

5 Nonlinear elliptic variational inequalities Preliminaries In order to explain the importance of elliptic variational inequalities, first consider the weak solution of the linear ...
3
votes
0answers
78 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
1
vote
2answers
95 views

Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
1
vote
1answer
81 views

On the density of $\mathcal{C}^\infty(\Omega) \cap W^{1,\infty}(\Omega)$ in $W^{1,\infty}(\Omega)$

Let $\Omega$ be an open set of $\mathbb{R}^n$ ($n\ge 1$). We know that the Meyers-Serrin theorem isn't true in $W^{1,\infty}(\Omega)$. But is it true that $\mathcal{C}^\infty(\Omega) \cap ...
2
votes
1answer
73 views

Characterization of Sobolev Space

I have just started learning about Sobolev spaces. So this might be trivial. I am working through the book "Partial Differential Equations" by Lawrence Evans, it came highly recommended. Taking ...
3
votes
1answer
64 views

Elliptic PDEs in Banach space

The standard textbooks discuss weak solutions and regularities in Hilbert spaces $W^{k,2}.$ I could not find a good reference on the theory based on Banach spaces $W^{k,p}.$ It would be good to point ...
1
vote
1answer
42 views

Proposed proof for Sobolev space result

I have the following result which seems that it must be true, but I would like to prove it: This is my proposed proof. If $U \subset \mathbb{R}^{n}$. Given $u \in W^{1,p}(U)$, where $u$ has compact ...
0
votes
0answers
14 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
2
votes
0answers
31 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
3
votes
1answer
70 views

A literature reference for Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth Riemannian manifolds

Anyone know a respectable reliable reference for the definition of Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth compact Riemannian manifolds. It suffices for m natural and $p\geq 1$
2
votes
1answer
33 views

how to prove that this weak solution is subharmonic?

My question is about this article http://hal.inria.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf. My question is : Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = ...
4
votes
0answers
79 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
0
votes
1answer
30 views

Reference needed for: $u \in H^1(0,T;L^2)$ if and only if $\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$

There is a result of the form: a function $u \in H^1(0,T;L^2)$ if and only if $$\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$$ holds for all $h \in [0,T]$. I have only seen one place ...
0
votes
0answers
104 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
0
votes
1answer
74 views

Compactness of Sobolev Space in L infinty

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can't directly use the compactness of Rellich Kondrachov ...
2
votes
0answers
57 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
0
votes
0answers
26 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
1
vote
1answer
36 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
3
votes
1answer
48 views

Sobolev spaces documentation

Can someone indicate some documentation on this subject? (thoroughly explained with a presentation of its applications - mostly interested in the FEM. However just a good presentation of the Sobolev ...
0
votes
0answers
56 views

Sobolev spaces in polar coordinates

I need some properties about Sobolev spaces in polar coordinates. To be precise, let $U = \{(x,y)\in\mathbb R^2 : x^2 + y^2 < R\}$ be an open disc and let $H_0^1(U)$ be the usual Sobolev space ...
2
votes
1answer
100 views

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...
1
vote
0answers
20 views

Characteristic Method for linear equation

I'm looking here for a reference to solve the following equation with the method of characteristic $$ \partial_t U + A\partial_x U = f$$ $$U(t=0,x)=U_0(x)$$ $$B(t,x=0)U = 0$$ where $(t,x)\in ...
7
votes
2answers
189 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
5
votes
2answers
127 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
0
votes
0answers
42 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
2
votes
2answers
100 views

The Poisson problem $-\Delta u =0$ with $u=g$ on the boundary where $g \in H^{\frac{1}{2}}$

Consider $$ \begin{align} -\Delta u =0 & \text{on $\Omega$} \\ u = g & \text{on $\partial\Omega$} \end{align} $$ where $g \in H^{\frac{1}{2}}(\partial\Omega)$. It seems there exists a ...
2
votes
0answers
25 views

Time continuity of a solution to the generalized porous medium equation

Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain. Now assume there is a weak ...
1
vote
2answers
82 views

Book searching in Elliptic Equation

I am learning a course with the subject of Elliptic Equations. If you know about it, please recommend me a book on Elliptic Equations. And if that's possible, someone post these books/author/...that ...
3
votes
1answer
216 views

What are degenerate PDEs and why weighted Sobolev spaces are useful?

As the title says, what exactly is a degenerate pde? I think it's a pde where it has a coefficient which shoots off to infinity at a point. Why is it that people use weighted Sobolev spaces to show ...
2
votes
2answers
134 views

Where to start to learn about free boundary problems?

I am interested in analysis of PDEs in free boundary problems. Please recommend some sources for me to learn about this. I would prefer Sobolev space setting as opposed to Holder spaces.
3
votes
1answer
143 views

Approximation of functions in fractional Sobolev spaces

First, some background and motivation: Convergence estimates in finite elements are often of the form $\|u-u_h \|_{L^2} \leq Ch^m \|u\|_{H^m}$, where $h$ is the mesh norm, $u_h$ is some discrete ...
4
votes
1answer
167 views

Blow up of solutions to parabolic PDE

I am looking for a text or answer detailing the blowup of solutions to parabolic PDE (eg. heat equation) in Sobolev space setting. I heard blowup is related to size of domain but I can't find any nice ...
3
votes
1answer
56 views

Nonlinear parabolic PDEs, what methods/techniques for existence?

I am curious what kinds of techniques one uses to show existence of PDEs with nonlinearities. I am aware of: 1) Minimisation problems 2) Semigroup (both of which I'd like to avoid) For linear ...
6
votes
1answer
138 views

Caccioppoli-Leray Inequality for De Giorgi's regularity theorem

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
1
vote
0answers
50 views

Multipliers in trace spaces

I need a reference for the following fact. Let $\Omega \subset \mathbb R^n$ be an open domain with $C^{1,1}$ boundary (maybe, less regularity is needed). Let $H^{1/2}(\Gamma)$ be the trace space of ...
4
votes
1answer
89 views

$\Delta u = \operatorname{div}f \ \ \mbox{in} \ \ B_1, f \in L^2 \Rightarrow \nabla u \in L^2$

I'm looking for results like, If $f \in L^p$ and $$ \begin{array}{rclcl} \Delta u & = & \operatorname{div}f & \mbox{in} & B_1\\ u&=&0& \mbox{on}& \partial B_1 ...
1
vote
1answer
65 views

Upper Bound of Sobolev norm by $L_2$ norm

A Paper by Madych and Potter states that if a function $f\in W_2^k(\mathbb{R})$ has evenly spaced zeroes (i.e. if $Z(f):=\{x:f(x)=0\}$, is such that $\underset{y\in\mathbb{R}}\sup ...
2
votes
2answers
116 views

Improvement of $W^{1,p}$ regularity of a elliptic equation solution.

I'm looking for some reference for results like $$ \mbox{div}(A(x) \nabla u) = 0, \ \ u \in H^1=W^{1,2} \Rightarrow u \in W^{1,p}, p>2 $$ where $A(x)$ is elliptic, this is, $Id\lambda \le A(x) \le ...
2
votes
0answers
66 views

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
7
votes
1answer
225 views

Short and elegant introduction to Sobolev spaces

I am preparing a course on Nonlinear Analysis, and I need to teach the most important facts about Sobolev spaces to my students. I know most books on this subject, from Brezis' to Adams', from Mazya's ...
3
votes
0answers
37 views

Unusual Compact Embeddings

Can anybody give a reference to the following two facts? The embeddings $$H_0^{1,2}(\mathbb R^n)\to L^2(\partial B_1(0))$$ and $$H^{1,2}(\partial B_1(0))\to H^{1/2,2}(\partial B_1(0))$$ are compact? ...
3
votes
1answer
166 views

Invariance under diffeomorphisms of the Sobolev $H^s$ spaces

Let $\Omega, \Omega' \subset \mathbb{R}^n$ be two open subsets and let $\psi : \Omega \rightarrow \Omega'$ be a $C^k$ diffeomorphism. Then, $\psi$ induces by pullback a linear isomorphism $$u \mapsto ...
3
votes
1answer
152 views

Time dependent or Bochner space references?

Does anyone have any recommendations where I can learn about time dependent or Bochner spaces? I mean spaces like $L^p(0,T; H^{-1}(\Omega))$. I think one needs some knowledge of distributions, so any ...
1
vote
1answer
211 views

Discrete Sobolev Space and Sobolev Spaces of Banach Space valued functions

This is a reference request. Can someone kindly give me some refernce(Books/papers) on Discrete Sobolev Space (like we use Discrete $L^p$ spaces of $g\colon\Omega\to\Bbb R $ maps with norm given as ...
1
vote
1answer
343 views

Reference of Sobolev space

currently I'm using Krylov's book, while consulting Evans (too many details are left out, for my level). Also, Adams 1975 version has been widely cited. So besides these ones, which book in your ...