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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.
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 for
In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is un
We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
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 Poincar
In this note, we improved the Liouville type theorem for the Beltrami flows. Two different methods are used to prove it. One is the monotonicity method, and the other is proof by contradiction. The conditions that we proposed on Beltrami flows are si