# Tagged Questions

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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### Question about boundedness of a sequence in $W^{3,q}$ for any $1\leq q < \frac{N}{N-1}$

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-...
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### Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} It is well known that if $f$ is ...
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### 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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### Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ for some $\overline{p}>p*'$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
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Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE u_t = \Delta F(u) \... 0answers 96 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 L^... 0answers 132 views ### If u has a weak derivative and f is C^1 does fu have a weak derivative (fractional Sobolev space and weak time derivatives) Let \Omega be an open bounded set. Let s \in (0,1) and H^s(\Omega) := W^{s,2}(\Omega). Let f \in C^1([0,T]\times \Omega) and u \in L^2(0,T;H^s(\Omega)) with weak derivative u' \in L^2(0,T;... 0answers 134 views ### Regularity theorem for Laplacian Let \Omega \subset \mathbb R^d be a bounded domain, d>2. Let f \in C^\infty(\Omega). If u \in L^2 is a distributional solution of \Delta u = f in \Omega then u \in C^\infty(\Omega) ... 0answers 158 views ### Show that the following u\in L^{\infty}\cap H^1(B) is a weak solution to the given system. Let B=B_{\exp(-2)}\subset\mathbb{R}^2. I would like to show that a weak solution to the following system (in B): \begin{align*} \triangle u_1&=-2|Du|^2(u_1+u_2)/(1+|u|^2)\\ \triangle u_2&... 0answers 122 views ### Proving norm equivalence in W^{1-1/p,p}(\Omega) Define for p\in [1,\infty) and \Omega=(0,1)^N\subset\mathbb{R}^NW^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$... 0answers 192 views ### Existence of weak solution in Sobolev space W^{1,p} Let B\colon W^{1,p}\times W^{1,q}\rightarrow \mathbb{R} and \frac{1}{p}+\frac{1}{q}=1 , p>2 where$$B[u,v]=\int \left( Du \,Dv+ h u v \right)\, dx$$where h>0. I am interested to prove ... 0answers 91 views ### What is H^1([0,1]) \otimes H^1([0,1])? Let H^1([0,1]) denote the Sobolev space H^1 on the interval [0,1]. What is H^1([0,1]) \otimes H^1([0,1])? Here, \otimes the tensor product of Hilbert spaces. In particular, how is that ... 0answers 451 views ### Showing that smoothing operators are compact Suppose I have a bounded, linear map T: H^1(X) \to H^1(X) such that T(H^1(X)) \subset C^\infty(X). Is T a compact operator? I'm guessing this depends on whether or not X is (pre)compact, and ... 0answers 101 views ### If F∈H^{-1}(Ω,ℝ^d) and ∃p∈\mathcal D'(Ω):\left.F\right|_{\mathcal D(Ω,ℝ^d)}=∇p, then ∃\overline p∈H^{-1}(Ω):F=∇\overline p Let d\in\mathbb N \Omega\subseteq\mathbb R^d be open \mathcal D(\Omega):=C_c^\infty(\Omega) and \mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d) H^{-1}(\Omega):=H_0^1(\Omega)'... 0answers 23 views ### Counterexample for the density of smooth functions in Sobolev spaces on a manifold I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds M,N the space C^\infty (M,N) is dense in L^p (M,N) if \dim(M) > p. (The ... 0answers 36 views ### Steps in alternative proof that if u \in H^1(\Omega), then Du = 0 a.e. on set \{u = 0\} Let \Omega be an open subset of \mathbb{R}^n, and let \phi be a smooth, bounded and nondecreasing function, such that \phi' is bounded and \phi(z) = z if |z| \le 1. Set$$u^\epsilon(x) := \...
How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...