# Tagged Questions

A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

189 views

165 views

### Nonlocal Stefan problems

Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality? A search on Google and MathSciNet give me only a handful of results which greatly ...
217 views

### Applications of composition operators on Sobolev spaces

I wold like to know some examples where composition operators on Sobolev spaces are useful. I'm in the following situation. $L^1_p(D)$ - homogeneous Sobolev space, in other words space of locally ...
186 views

### orthonormal basis or Parseval frame for Sobolev spaces

Consider the uni-variate Sobolev space of order $m$: $$\mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}.$$ It is ...
124 views

### Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
91 views

229 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 ...
280 views

### Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in sup-norm)....
135 views

### convergence of $e^{it\Delta}f$

I heard of a conjecture that $e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$ but couldn't find a proper reference.
169 views

### Weak solution of a heat equation is zero?

I work on a bounded domain in $\mathbb{R}^n$. Let $u \in H^1(0,T;H^{-1})\cap L^2(0,T;H^1)$ be a solution of the heat equation: $$\langle u', v \rangle + \int \nabla u \nabla v = 0$$ for each test ...
374 views

### How do functions operate in a Sobolev space $H^{s}$?

Let $s>\frac{1}{2};$ and define a Sobolev space as follows: $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ Fact: Let $m$ ...
135 views

### Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact. Is it true that the embedding $H^1_0(\Omega) \rightarrow L^2_K(\Omega)$...
192 views

174 views

279 views

294 views

485 views

First, I give my motivation to ask this question. The generalised Neumann trace can be defined as $${}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial u}{\partial{\mathbf{n}}},v\rangle_{H^{1/2}(\... 0answers 118 views ### Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs) Denote by \mathbb{E}(g) the solution of the PDE$$\Delta v(x,y) = 0 \quad\text{in $\Omega$}v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$Let X=L^1(\partial\Omega). Let h(t) be a ... 2answers 381 views ### A question on density of Lipschitz functions in weighted Sobolev spaces Recall that for a domain \Omega\subset \mathbb{R}^n, the weighted Sobolev space W^{1,n}(\Omega,\mu) is defined as f\in L^n(\Omega,\mu) and the weak derivative Df\in L^n(\Omega,\mu). Let now ... 1answer 81 views ### Coercivity for functional and complete orthonormal system Consider with \rho \in W^{1,2}([0,\pi]) the following functional$$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$I know that in the L^{2}([0,\pi]) the coercivity condition is satisfied, but i'm ... 1answer 742 views ### A comparison principle for parabolic equation (Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear) Suppose F:\mathbb{R} \to \mathbb{R} is smooth with F(x) > 0 for ... 1answer 283 views ### reference request: trace/lifting operator for L^{\infty} data in bounded \Omega\subset R^d I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ... 0answers 76 views ### Is \partial^\alpha a map H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)? More precisely, the question is formulated as follows. Let F be an arbitrary Banach space, especially not having the UMD−property. Let N\in\mathbb N and s\in\mathbb R and 1\le p\le +\infty . ... 1answer 817 views ### For which maps S^1\to S^1 is the winding number defined? There are two classes of maps S^1\to S^1 for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map \mathbb R\to S^... 1answer 145 views ### Getting an a priori bound on a nonlinear gradient term in PDE; how to adapt trick from L^2 case to H^{-1} case? I have the PDE$$u_t(t) - \Delta f(u(t)) = 0$$in H^{-1}(\Omega) where f is a nonlinear function. Define F(s) = \int_0^s f(s). Note that if u_t(t) \in L^2(\Omega),$$\frac{d}{dt}F(u(t)) = f(...
Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...