# Questions tagged [sobolev-spaces]

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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### $L^\infty$ boundedness for solution of elliptic PDE with Neumann BC

On a bounded smooth domain in $\mathbb{R}^n$ consider the equation $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$where $k>0$ is a constant and $f \in L^2(\Omega)$. We know that $u \in H^2(\Omega)$. ...
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### Direct sum of Sobolev spaces

We know that we may decompose $L^2(\mathbb{R})$ as $L^2(\mathbb{R^-})\oplus L^2(\mathbb{R^+})$. Can we write $L^2(\mathbb{R})=L^2(\mathbb{R_-^*})\oplus L^2(\mathbb{R_+^*})$? Now, let $H^1(\mathbb{R})$...
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### Question on trace Sobolev's theorem for domain $\Omega \times (0,T)$

Let $\Omega \subset \mathbb R^3$ be an open,bounded subset with a $C^2-$boundary $\Gamma$. Fix $T>0$. Can we claim that $W^{1,2}(\Omega \times (0,T)) \hookrightarrow C([0,T];L^2(\Gamma))(*)$ ...
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### Convergence of $f(u_n) \to f(u)$ when $u_n \to u$ in $L^p$/Sobolev spaces

I got a function $f\colon \mathbb{R} \to \mathbb{R}$ which is such that $0 \leq f \leq 1$ and it is a smooth function. Suppose $u_n \to u$ in $H^1(\Omega)$. $\Omega$ is a bounded smooth domain What ...
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### Conditions on the domain for Sobolev embeddings

I am reading the proof of the Sobolev embedding theorem presented in the book Sobolev Spaces by Robert A. Adams and John J. F. Fournier. I could not understand the proof for part II of the theorem. ...
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I don't understand why this inequality holds (i found it in some notes): Let $v\in L^1 \cap C_c,$ then $$\Vert v\Vert_{L^{q}} \leq \Vert v\Vert_{L^{1}}^{\frac{1}{q}} \cdot \Vert v\Vert_{L^{\infty}}... 0answers 24 views ### To show a function is in a Sobolev space, can we use weak spherical derivatives? For example, say \Omega=B(0,1) in \mathbb{R^2}, and I have a function represented by$$u(x)=\begin{cases}\ln\ln1/|x|& |x|\leq e^{-2}\\\ln(2) &e^{-2}<|x|\leq 1.\end{cases}$$There is a ... 0answers 27 views ### References on equivalent characterization for Sobolev spaces of functions of one variable I cited a result which characterizes Sobolev spaces of functions of one variable as$$H^p(a,b):=\{x\in C^{p-1}[a,b]:x^{(p-1)}(t)=α+\int^t_aΨ\,\mathrm ds,\ α\in\mathbb R,\ Ψ\in L^2\},$$where p\in\... 0answers 41 views ### Can I pass to the limit? Let be a sequence u_n of C_0^\infty (\mathbb{R}^N)-functions converging to u in H^1(\mathbb{R}^N), which implies that u_n\rightarrow u in L^2(\mathbb{R}^N) and \nabla u_n\rightarrow \... 1answer 76 views ### Integration by parts for u \in H^{1} and v\in H^{1}_{0} Let \Omega be a smoothly, open bounded domain in \mathbb{R}^{n}. Assume that u\in W^{1,2}\left(\Omega\right) and v\in W^{1,2}_{0}\left(\Omega\right). Is the integration by parts always true, ... 1answer 59 views ###  f\in C_{0}^{\infty} \Rightarrow f\in L^m ? Let  C_{0}^{\infty}  be the subspace of  C^{\infty}  functions with compact support in  R^n . It's true that if  f \in C_{0}^{\infty},  then  f \in L^{m} \cap H^{s}  where  1\leq m \leq 2 ... 1answer 26 views ### Norm in Sobolev Spaces. In an exercise, i would like to use the next afirmation. Si L=\delta_0 y v\in H^1_0 entonces |<L,v>| =|v(0)| \le ||v||_\infty \le K_1 ||v||_{H^1} \le K_2 ||v||_{H^1_0}. But, i do not ... 1answer 73 views ### Is this Poincaré-type inequality valid? It is proved that the Poincaré inequality is still true for functions with zero mean boundary traces. Motivated by this, I have the following question: Let \Omega be an open,bounded and connected ... 1answer 83 views ### What Soboblev embedding i could use in this case? I'm trying to understand why this inequality is true$$ \Vert f\Vert_{L^2}^{p}\cdot h\left(\Vert f\Vert_{\infty}\right) \leq c \Vert f\Vert_{k+1}^{p} \cdot h\left(\Vert f\Vert_{k+1}\right), $$where ... 1answer 86 views ### Gauss - Green theorem for Sobolev H^1 space I know the Gauss-Green theorem: Let U \subset \mathbb{R}^n be an open, bounded set with ∂U being C^1. Suppose u ∈ C^1(\bar U), then$$∫_U u_{x_i} dx = \int_{∂U} u \nu^i dS,$$where \nu=(\... 0answers 33 views ### Compactness of Sobolev Space Let \Omega \subset \mathbb{R} be a bounded domain and 2<p<\infty. Assume u \in C^{2,1}(\Omega \times (0,\infty))\cap C^{1}((0,\infty);L^{2}(\Omega))\cap C([0,\infty);H_{0}^{1}(\Omega)) ... 1answer 47 views ### Extension of weak derivatives in Bochner spaces I am struggling to understand estimate (15) from the following proof from the PDE book by Evans: He argues that estimate (15) follows from difference quotients, but I can't understand this. In ... 0answers 30 views ### Limit cases in sobolev embedding I am trying to find coumterexamples for the critic cases in sobolev embedding theorem for all \mathbb{R}^N. For instance, a function u\in W^{1,p}(\mathbb{R}^N) s.t. u\notin L^q(\mathbb{R}^N) for ... 1answer 18 views ### Existence of convergent sequence implies convergence of function as t\to\infty Let \Omega \subset \mathbb{R} be an unbounded domain and u \in C([0,\infty);H_{0}^{1}(\Omega)). Assume there exists a sequence \{t_{n}\}_{n\in\mathbb{N}} such that t_{n}\to\infty and ||u(t_{n}... 1answer 30 views ### Isometric embedding of L^2 onto H^{-1} Let X be a Banach space. Many sources in the literature identify L^2(X) with H^{-1}(X) through the identification$$ \varphi: L^2(X) \to H^{-1}(X); \quad \quad \varphi(u)(v) := (u,v)_{L^2}, \...
With inputs $X_1, \dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $\hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of ...