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Register a new userAn Improved Compact Embedding Theorem for Degenerate Sobolev Spaces
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This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $Omega$ with respect to the norm: $$|f|_{QH^{1,p}(v,mu;Omega)} = |f|_{L^p_v(Omega)} + | abla f|_{mathcal{L}^p_Q(mu;Omega)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $Omega$. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition of the form $$w(x)|xi|^p leq left(xicdot Q(x)xiright)^{p/2}leq tau(x)|xi|^p$$ for a pair of $p$-admissible weights $wleq tau$ in $Omega$. We also give explicit examples demonstrating the sharpness of our hypotheses.
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We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic forms. Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegener- ate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains, the latter for weights equal to powers of the distance to the boundary. We also derive a compactness result for Lebesgue spaces on quasimetric spaces unrelated to Euclidean space and possibly without any notion of gradient.
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The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $RR^n$. The analysis interestingly combines use of Poincare inequalities and of some Hardy type inequalities.
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