# 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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### Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of $R^n$....
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### 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 $\Omega \subseteq \mathbb{R}^n$. ...
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### Sobolev Spaces separable

How do I demonstrate that the Sobolev spaces $W^{1,\infty}$ is not separable? PS: I know that space $L^{1,\infty}$ is not separable but was unable to use this information.
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### $(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$$ where $(g,f)_{L^2}$ is inner product and $f$ and $g$ have higher enough regularity ...
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### 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 ...
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### 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 ...
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### Extending $W^{2,n}(\Omega)\cap C^2(\Omega)$-functions to $W^{2,n}(\Omega)$

I have the following theorem: Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is ...
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### Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
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### “Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
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### A question on weakly convergence and norm convergence.

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
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### A question on a bounded sequence in $H^1(\mathbb{R}^n)$.

Let $r>0$, and $2 \le q \le 2^*$. Suppose that $\{u_k\}_k$ is a bounded sequence in $H^1(\mathbb{R}^n)$ and $\lim_{k\to \infty} \sup_{y \in \mathbb{R}^n} \int_{B_r(y)}|u_k|^q dx \rightarrow 0.$ ...
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### Sobolev spaces in $\mathbb R^n$ with functions having support on a closed set

I am interested in $H^s$ Sobolev spaces in $\mathbb R^n$ which have functions with support in a given closed set $K$ , denoted by $H^s_K$. Here $K$ is the complement of some bounded open set $\Omega$. ...
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Let $\Sigma, M$ be compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by W^{k,p}(\Sigma,M) = \{ u \in W^{k,p}(\... 0answers 83 views ### ODE with irregular coefficient Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution h(x) to be (at least) continuous with its first and second order derivative exist only ... 0answers 201 views ### An exercise about Sobolev Spaces Let \Omega\subset\mathbb{R}^n be an limited open set of class C^1 and 1\leq p<\infty. Show that\bigcap_{m=0}^{\infty}W^{m,p}(\Omega)=C^{\infty}(\overline{\Omega}).$$0answers 135 views ### How can we glue Sobolev functions? Let u:A\cup B\to R be a function (where A and B are disjoint connected sets and A\cup B is connected) such that u restrict to A and to B are in W^{1, p}. Which result guarantees me ... 0answers 380 views ### Weak convergence and limit of this sequence Let f_n be bounded uniformly in the H^1 norm, so we have (weak convergence)$$f_n \rightharpoonup f \qquad \text{in} \qquad H^1(\Omega\times [0,T]).$$Then by compact embedding, we have strong ... 0answers 105 views ### Compactness of an embedding between weighted spaces I read somewhere that if: N\geq 2 is an integer, p\in ]1,N[, r>N/p, m\in L^r(0,a) (with a>0) and m>0 a.e. in (0,a), then the weighted Sobolev space W^{1,p^\prime} ((0,a),m^{... 0answers 82 views ### Difference between C^\infty (U) and C^\infty (\overline U) I am learning Sobolev spaces. There seem to be a difference while approximating a function in W^{k,P}(\Omega) by smooth function C^\infty (\Omega) and C^\infty ( \overline \Omega), where we use ... 0answers 89 views ### Extension of Uncertainty Relations to a specific potential in Schrödinger Equation Given some \|\psi \| \in L^2 (\mathbb R^n)  such that \| \psi \|_2 =1 and a function (potential) V: \mathbb R^n \rightarrow \mathbb R. The SchorĂ¶dinger equation tells us that -\triangle \... 0answers 324 views ### Adjoint of multiplication operator on Sobolev space This is sort of an idle question, and I'll admit I didn't think very hard about it. Let H^1 = H^1(\mathbb{R}^n) be the Sobolev space with norm ||f||_{H^1}^2 = ||f||_{L^2(\mathbb{R}^n)}^2 + ||\... 0answers 20 views ### I don't see why W^{1, 2}(\partial D) being compactly embedded in L^2(\partial D) lets us show an operator is Fredholm of index zero. Let D be a bounded Lipschitz domain. Let A be the single layer potential which maps L^2(\partial D) into W^{1, 2}(\partial D) boundedly. A is given by:$$ A_D[\phi] = \int_{\partial D}G(x-y)...
Let $\Omega\subset \mathbb R^2$ be given such that $\Omega$ is open bounded with nice boundary. Let $\Gamma$ be a finite segment in $\Omega$. We also define $\Omega_1:=\Omega\setminus \Gamma$. ...
Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) \$\mathcal ...