# 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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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 $$\int_{\partial\Omega} |u|^2 ds \... 0answers 64 views +50 ### Global Euclidean Carleman Estimate with a linear phase I am interested in deriving the following global Carleman estimate which I think should hold :  \| e^{\tau \phi} \triangle e^{-\tau \phi} u \|_{L_{\delta+1}^2({\mathbb{R^3})}}> C \tau \| u \|_{L^... 0answers 59 views ### Modify the jump set of BV function Let u\in BV(\Omega) be a function of bounded variation where \Omega\subset \mathbb R^N is open bounded with smooth boundary. We use Du to denote the weak derivative of u. (So Du is a Radon ... 1answer 65 views ### What is the function space H^1_{m, \sigma}? I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order". Theorem 3.1 is the first time in their paper that the function space ... 1answer 197 views ### Simplicity of eigenvalues Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where p \in C([0, 1]), p \ge \alpha > 0 on I, and q \in C([0, 1]). No further assumptions are made; in ... 0answers 52 views ### Sobolev spaces defined on non-compact Lie groups In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ... 1answer 100 views ### If u \in H^1(U), then Du = 0 almost everywhere on the set \{u = 0\}, auxiliary result Let \phi be a smooth, bounded and nondecreasing function, such that \phi' is bounded and \phi(z) = z if |z| \le 1. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have that... 0answers 48 views ### Continuous inclusions Sobolev theorem, question [closed] How do I see that if f, g \in H^s(\mathbb{R}^n) for s > n/2, then fg \in H^s(\mathbb{R}^n) and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ... 1answer 81 views ### L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X) for X \subset Y dense? is the Inclusion stated in the title true? In my case the spaces (essentially) are X = H^1(\Omega) and Y = L^2(\Omega) for \Omega \subset \mathbb{R} bounded. My first try was to show \lim_{t_1 ... 3answers 147 views ### Exists C = C(\epsilon, q) such that \|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} for all W^{1, 1}(0, 1)? [closed] Let 1 \le p < \infty. For all \epsilon > 0, does there exist C = C(\epsilon, q) such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$. Suppose that \sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{...