# 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.

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### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $R^d$. (Say, a ball, say a cube...) For which classes $\cal C$ of functions, every function $f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
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### $W^{1, p}(M, N)$ path-connected if and only if $C^0(M, N)$ is path-connected

I'm asked to show that for compact, smooth Riemmanian manifolds $M$, $N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from "...
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### Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
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I am interested in deriving the following global Carleman estimate which I think should hold : $\| e^{\tau \phi} \triangle e^{-\tau \phi} u \|_{L_{\delta+1}^2({\mathbb{R^3})}}> C \tau \| u \|_{L^... 0answers 59 views ### Modify the jump set of$BV$function Let$u\in BV(\Omega)$be a function of bounded variation where$\Omega\subset \mathbb R^N$is open bounded with smooth boundary. We use$Du$to denote the weak derivative of$u$. (So$Du$is a Radon ... 0answers 107 views ### Completion of$C_{0,rad}^{\infty}(\Omega)$with respect to the norm$\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $I have a question that it seems simple but I can not solve it. Let$\Omega$be the unit ball centered at zero in$\mathbb{R}^N$,$N>4$. Assume that$C_{0,rad}^{\infty}(\Omega)$is the space of all ... 0answers 59 views ### Traces of fractional Sobolev spaces$W^{s,p}$with$0<s<1/p$I've stumbled upon a problem involving the trace of a function in a fractional Sobolev space of the form$W^{s,2}(H)$, where$H$is a half-plane in$\mathbb{R}^2$. Would it be possible to define a ... 0answers 114 views ### Compact embedding of${\rm L}^1_{loc}$space I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let$\Omega\subset{\bf R}^2$be open and bounded and let$p\in\langle 1,2\rangle$. ... 0answers 53 views ### Are functions whose partial derivatives are simple functions dense in$W^{1,\infty}$? In a 2D domain, are the functions whose partial derivatives are simple functions dense in$W^{1,\infty}$? 0answers 59 views ### Pointwise (a.e) evaluation of$\sum_{n \geq 0}(u,w_n)_{L^2}w_n$and equalities in$L^2$Let$w_n$be a orthonormal basis of$L^2(\Omega)$. Given$u \in L^2$we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$ Suppose$w_nare the eigenfunctions of the Neumann Laplacian. We can write $$\... 0answers 171 views ### Lipschitz continuity of a composition operator Let M be a compact Riemannian submanifold of \mathbb{R}^K, U\subset \mathbb{R}^K an open neighboorhood of M such that the shortest point Projection P_M\colon U\rightarrow M is well-defined ... 0answers 157 views ### Bounded functions dense in Sobolev Spaces Let M be a complete Riemannian manifold. Is it always true that the subspace C^2_b(M)\cap W^{2,p}(M) is dense in W^{2, p}(M), where C^2_b(M) denotes the space of functions that are uniformly ... 0answers 171 views ### Strong solution to u_t - \Delta_p u = f For p > 1, consider the equation$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangleu(0) = u_0u|_{\partial\Omega} =0for all v \in W^{1,p}(... 0answers 224 views ### Regularity of 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 =... 0answers 164 views ### Linear interpolation in weighted Sobolev spaces I am reading a paper regarding the pricing of certain financial derivatives making use of the finite element method. In this paper the following weighted Sobolev spaces are introduced: W_{0} = \{ ... 0answers 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. 0answers 105 views ### Existence and smoothness of convolutions of distributions in Sobolev spaces Let f\in H^{s_1}(\mathbb{R}^n) and g\in H^{s_2}(\mathbb{R}^n), where s_1, s_2 \in \mathbb{R} and can be positive or negative. It is easy to show that f *g is defined pointwise when s_1+s_2\... 0answers 189 views ### Degree of a function in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1) We can define the degree of a function f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1) as\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial \... 0answers 200 views ###\tilde{u}=0,\ a.e.\ x\in\Gamma$? Suppose$\Omega\subset\mathbb{R}^n$is a bounded domain. Let$u\in H_0^1(\Omega)$and$\tilde{u}\in L^2(\Omega)$satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in C_0^{\... 0answers 518 views ### Open sets and Poincaré's inequality In many references, Poincaré inequality is presented in the following way : Let \Omega\subset \mathbb R^d an open bounded set. We can find a constant C which depend of \Omega such that for ... 0answers 225 views ### Where can I find interpolation inequalities for derivatives of the following form? Here they are:$$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$and$$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ... 0answers 57 views ### A specific mollified functions in the Sobolev space H^1(R) Let$u>0$be in$H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of$C^{\infty}$functions with compact support are dense in the Sobolev space$H^{1}(\mathbb{R})$. Hence, we have a ... 0answers 79 views ### Getting an a priori energy estimate from PDE weak formulation On a bounded domain$\Omega$, I have two functions$u$and$v$in$L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$satisfying $$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv... 0answers 50 views ### Constant in a trace Sobolev theorem for concave domains I wonder is the following inequality is true/known: Let \Omega\subset \mathbb{R}^n be a (locally) Lipschitz domain which is the complement of a convex set, then$$ \int_{\partial\Omega} |u|^2 ds \... 0answers 157 views ###$L^\infty$bound on solutions of linear parabolic equations We work on a closed Riemannian manifold$M$. Let$u$and$v$be the non-negative weak solutions of $$au_t - 2d\,\Delta au = cv - f$$ $$bv_t - d\,\Delta bv = f$$ $$u(0)=u_0, \quad v(0)=v_0$$ where$f$... 0answers 75 views ### Multiplier operators on anisotropic weighted$L^2$spaces Suppose$\mathcal{M}$is a multiplier operator on$L^2(\mathbb{R})$, in the sense that, for any$u(x)\in L^2(\mathbb{R})$,$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$where the scalar complex function ... 0answers 131 views ###$L^\infty-L^1$smoothing effect for the heat equation Let$\Omega$be a bounded domain in$\mathbb{R}^n$. Let$u \in L^2(0,T;V)$be the weak solution of the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ where$u_0$is bounded initial data. Here ... 0answers 51 views ### Doubts regarding pre-compactness of bounded sequence of measure valued functions Given:$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$be a Caratheodory vector such that for each$M \gt 0 $,$\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$. ... 0answers 40 views ### Trace space of$\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$for$s \in (-\frac 12, \frac 12)$Let$s \in (-\frac 12,\frac 12)$and let$X=D(\Lambda)$be a Hilbert space with$\Lambda$the infinitesimal generator of a bounded semigroup of class$C^0$in$Y$(which is another Hilbert space), and ... 0answers 89 views ### Rellich Embedding Theorem for the$2$-Sphere I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the$2$-sphere. To be precise, for$S$the spinor bundle of$S^2$;$L^2(S^2)$the space of square integrable ... 0answers 79 views ### elliptic regularity for Neumann BVP on square I am interested in the regularity of ellitpic equations like $$-\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with$ \partial_\nu u =0$on$ \partial \Omega$where$ \Omega=(...
Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$...