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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Is this integration by parts legitimate?

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,$$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
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Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?

Define $$Q = \bigcup_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required. Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
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Trace Theorem for $p=\infty$

I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one (...
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Define $W(X,Y) = \{ u \in L^2(0,T;X) \mid u' \in L^2(0,T;Y)\}$ where $u'$ is the usual weak derivative. Let $V \subset H \subset V^*$ be a Hilbert triple. If $u \in W(V,V)$ (so $$u \in L^2(0,T;V) \... 1answer 197 views L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)? Put, C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}(= Continuous functions on \mathbb R vanishing at \... 0answers 98 views Almost a Green formula Let \Omega be the half-space \mathbb{R}^{n-1}\times \{ x_n>0 \}, let v \in L^2(\Omega) and \phi\in \mathcal{C}^{\infty}(\overline{\Omega}) with compact support in \overline{\Omega}. What ... 1answer 110 views well-posedness of heat equation with Neumann BC and periodic data On a domain \Omega with f \in L^2(0,T;H^{-1}) such that f(0) = f(T), consider$$u_t - \Delta u = f\quad\text{on $\Omega$}\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$... 1answer 253 views Equivalence of negative Sobolev norm of derivative to L^2-norm Let S:=(0,1)^2 be the unit square in \mathbb{R}^2, and let M:=\{u\in L^2(S)\mid \int_S u=0\} be the space of (real-valued) L^2-functions with mean value zero. On M we can consider the L^2(S)... 0answers 139 views Weak periodic solution of parabolic PDE Take$$ u_t(t) + A(t)u(t) = f(t),  u(0) = u(T), $$where A is an linear elliptic operator and the first equation is an equality in L^2(0,T;V^*) for V \subset H \subset V^* Hilbert triple. (... 1answer 246 views Why can't I get global existence to linear PDE in this way? [closed] For any n > 0, standard theory implies there is a unique u_n \in L^2(0,n;V) with u_n' \in L^2(0,n;V^*) such that$$u_n' + Au_n = f\quad\text{as an equality in $L^2(0,n;V^*)$}u_n(0) = u_0$... 1answer 244 views Inequality in the Sobolev space$H^1$I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ^2\bigg)^{\frac{q}{2}-a}+\frac{c}{r^{2a}}\bigg(\int_{B_r}... 1answer 174 views A Poincaré-type inequality with logarithmic function For any function f(x) we denote \bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx. Let \Omega\subset \mathbb{R}^n be a bounded smooth domain and u(x)> 0 be a smooth function defined on \Omega.... 2answers 357 views Showing H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega) is continuous? Let \Omega\subset\mathbb R^n be a bounded Lipschitz domain. How does one prove that the inclusion H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega) is continuous? I define H^{\frac 1 2}(\... 2answers 113 views References for well-posedness of weak solutions to Stefan problem Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)? ... 2answers 245 views Interior Schauder estimates with weights Suppose we have u(x)\in H_2^{loc}(\Omega_{\rho}), where \Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}, and in \Omega_{\rho}, u satisfies the equation$$ \Delta u-V(x)u=0, $$where V is a ... 1answer 170 views If f \in H^{\frac 12} and \varphi is Lipschitz, is f\varphi \in H^{\frac 12} (on a Lipschitz hypersurface)? Let M be a bounded hypersurface. Let f \in H^{\frac 12}(M) and let \varphi\colon M \to \mathbb{R} be a Lipschitz function. When M=\Omega \subset \mathbb{R}^n an open domain, we know that the ... 2answers 367 views property of local sobolev space The local Sobolev space,defined as W^{k,p}_{loc}(\Omega), is the space such that for any u \in W^{k,p}_{loc}(\Omega) and any compact V\subset \Omega, u \in W^{k,p}(V). I am just wondering if ... 2answers 445 views Sobolev spaces on boundaries Consider the Sobolev space W^{s,2}=H^s for s=\frac{1}{2}. Let \Omega \subset \mathbb{R}^n be an open set with boundary \partial\Omega. I have seen two definitions of the space H^s(\partial\... 1answer 86 views Which rate of growth of the Sobolev norms guarantees analyticity? Let u\in C^\infty(\mathbb T^k), where \mathbb T^k is the k-dimensional torus. (Equivalently, u\in\mathbb R^k and u is 2\pi-periodic with respect of each argument.) We define the semi-norm ... 1answer 151 views sub and super-levelset regularity for Sobolev functions I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions u\in W^{1,p}(\mathbb{R}^d). More precisely: Assume ... 0answers 223 views Alternative representations of Sobolev space Is there a way to represent a Sobolev space as the image of a fractional integral operator over an L^p Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ... 1answer 119 views Examples of optimal ultracontractivity estimates for a Markovian semigroup T_t that do not depend polynomialy on t Let (X,\mu) be a measure space and T_t : L_2(\mu) \to L_2(\mu) for t \geq 0 a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:$$ \| T_t : L_p(\mu) \to L_q(\mu)\| \... 1answer 410 views traces of sobolev spaces under additional assumptions Let$p\in [1,\infty]$,$\Omega$an open bounded domain with (smooth, if necessary) boundary$\partial \Omega$. Is there a subspace$X\subset L^p(\Omega)$- a simply describable space, ideally a ... 1answer 175 views Examples of functions in$W^{k,p}(\Omega)$with exact smoothness Please give, explicitly, a function$f:\Omega\mapsto\mathbb{R}$such that$f\in W^{k,p}(\Omega)$but$f\notin W^{s,p}(\Omega)$for$s>k$. Here$\Omega$can be a subset of$\mathbb{R}^n$with ... 1answer 207 views functions of bounded variation and gradient vector measure I want to prove a function of bounded variation on some domain$D\subset R^n$,$f\in BV(D)$, has the property that there is a constant$C$, such that $$\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} \int_{... 4answers 860 views Variation on the Sobolev space H^1_0 Let \Omega\subset\mathbb{R}^n be a bounded open set, let$$ C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\}, $$and let C^1_c(\Omega) be the space of ... 1answer 208 views Isocapacity inequalities in the theory of Sobolev Spaces Section 9 of the lectures notes of Maz'ya (Мазья) on isocapacity, lectures notes that can be found at: http://www.math.liu.se/~vlmaz/pdf/mazya.pdf, discusses inequality between the L^p norm in a ... 4answers 1k views When to use more exciting function spaces than ordinary Sobolev spaces? In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions. For example, Sobolev spaces L^2(0,T;H^... 1answer 241 views Compact embedding of weighted sobolev spaces in continous functions spaces Let the weighted sobolev space M^p_{s,\delta} be the completion of C_0^\infty(\mathbb{R}^n) in the norm \sum_{\left\vert\alpha\right\vert\leq s}\left\Vert(1+\left\vert x\right\... 0answers 189 views Degree of a function in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1) We can define the degree of a function f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1) as$$\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial \... 1answer 208 views Dependence of the constant in Korn's inequality on the domain Let$\Omega \subset \mathbb{R}^N$be an open, connected set with Lipschitz boundary,$N \geqslant 2$and $$\mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} (... 1answer 177 views Does a particular iteration produce a weak solution to a non linear pde? Consider the following non linear pde in the unknown v(x,y):$$ \frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$where t is some fixed small ... 0answers 192 views Is there an appropriate weighted Sobolev space to include exponential map and projection map? Observe that given a non negative function \omega: \mathbb{R^2} \rightarrow [0, \infty), we can define the weighted L^{p}(\mathbb{R}^2, \omega) spaces. They are measurable functions f: \... 0answers 302 views Moduli of smoothness, Besov spaces, and Sobolev spaces For 1\leq p\leq\infty, the r-th order L^p-modulus of smoothness is $$\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}$$ where \Omega_{rh}=\{... 0answers 447 views Classes of (non-continuous) functions with the fixed point property Let K be a convex body in R^d. (Say, a ball, say a cube...) For which classes \cal C of functions, every function f \in {\cal C} which takes K into itself admits a fixed point in K. ... 1answer 367 views Caccioppoli-Leray Inequality for De Giorgi's theorem proof I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ... 1answer 303 views about smoothing pseudodifferential operators Hi, I have a question which involves pdo. Let us consider a pseudodifferential operator A:S(\mathbb{R}^d)\rightarrow S(\mathbb{R}^d) whose symbol a(x,\xi) lives in the S_{0,0}^0 class :$$ \... 0answers 154 views compact embedding in duals of weighted Sobolev spaces On the whole space$\mathbb{R}^d$consider the weight$\omega(x)=\sqrt{1+|x|}$. Under which conditions on$k,q$is the embedding$$L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset (W^{k,q}(\mathbb{R}^d,\... 1answer 179 views Is BV2 space closed in L2 space? We define the BV2 space by$S = \lbrace f\in L^2:\textrm{TV}(f)<\infty\rbrace$, where$TV(f)=\sup_{g\in C_c^1,\|g\|_\infty\leq 1}\int f\cdot \textrm{div}g$. My question is: is$S$closed in$L^2$? ... 0answers 154 views Adjoint operator in sobolev space Let$g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$and let us define the operator$B : y \to g y$from$H:=H_0^1(\Omega)\cap H^2(\Omega)$to$H$, which we endowed with norm$|u|=(\|u\|^2 +\|\Delta u\|...
I already asked this on M.SE, but get no answers. Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not ...
Let $\Omega$ be a domain in $\mathbb{R}^3$, does there exist a vector-valued Sobolev-type space, or maybe space in other sense, $V$, satisfying the following: 1) \$S:=\lbrace v\in V:\|v\|_{L^\infty(\...