On trace theorems for weighted mixed-norm Sobolev spaces and applications
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Tuoc Phan
Abstract
We prove trace theorems for weighted, mixed-norm, Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They recover classical results in the case of unmixed-norm spaces. The work is motivated by the study of regularity theory for solutions of elliptic and parabolic equations with anisotropic features and with nonhomogeneous boundary conditions. The results provide an essential ingredient to the study of fractional elliptic and parabolic equations in divergence form with measurable coefficients.
Abstract
We prove trace theorems for weighted, mixed-norm, Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They recover classical results in the case of unmixed-norm spaces. The work is motivated by the study of regularity theory for solutions of elliptic and parabolic equations with anisotropic features and with nonhomogeneous boundary conditions. The results provide an essential ingredient to the study of fractional elliptic and parabolic equations in divergence form with measurable coefficients.
Kapitel in diesem Buch
- Frontmatter I
- Foreword V
- Contents VII
- Yau’s conjecture on the dimension of harmonic polynomials 1
- On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces 23
- On trace theorems for weighted mixed-norm Sobolev spaces and applications 49
- Sharp stability of the logarithmic Sobolev inequality in the critical point setting 77
- On a variational problem of nematic liquid crystal droplets 103
- Estimates for the variable order Riesz potential with applications 127
- A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces 157
- A Bliss–Adams inequality 179
- Trudinger-type inequalities in RN with radial increasing mass-weight 197
- In response to David R. Adams’ October 12, 2001, letter 215
- Some remarks on capacitary integrals and measure theory 235
- Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights 265
- Index 287
Kapitel in diesem Buch
- Frontmatter I
- Foreword V
- Contents VII
- Yau’s conjecture on the dimension of harmonic polynomials 1
- On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces 23
- On trace theorems for weighted mixed-norm Sobolev spaces and applications 49
- Sharp stability of the logarithmic Sobolev inequality in the critical point setting 77
- On a variational problem of nematic liquid crystal droplets 103
- Estimates for the variable order Riesz potential with applications 127
- A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaces 157
- A Bliss–Adams inequality 179
- Trudinger-type inequalities in RN with radial increasing mass-weight 197
- In response to David R. Adams’ October 12, 2001, letter 215
- Some remarks on capacitary integrals and measure theory 235
- Remarks on vector-valued Gagliardo and Poincaré–Sobolev-type inequalities with weights 265
- Index 287
