22 research outputs found
Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups
Let X_1,...,X_q be the basis of the space of horizontal vector fields on a
homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system
of N equations, in divergence form, structured on these vector fields, where
the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued
bounded measurable functions defined in a bounded domain A of R^n, satisfying
the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined
by the Carnot-Caratheodory distance induced by the X_i's). We prove interior
HW^{1,p} estimates (2<p<\infty) for weak solutions to the system
Local real analysis in locally homogeneous spaces
We introduce the concept of locally homogeneous space, and prove in this
context L^p and Holder estimates for singular and fractional integrals, as well
as L^p estimates on the commutator of a singular or fractional integral with a
BMO or VMO function. These results are motivated by local a-priori estimates
for subelliptic equations
Optimal concentration level of anisotropic Trudinger-Moser functionals on any bounded domain
Let be convex and homogeneous of degree , its polar represent
a finsler metric on , and be any bounded open set in
. In this paper, we first construct the theoretical structure
of anisotropic harmonic transplantation. Using the anisotropic harmonic
transplantation, co-area formula, limiting Sobolev approximation method,
delicate estimate of level set of Green function, we investigate the optimal
concentration level of the Trudinger-Moser functional under the anisotropic
Dirichlet norm constraint , where
denotes the sharp constant of anisotropic Trudinger-Moser inequality in bounded
domain and is the Lebesgue measure of the unit Wulff ball. As an
application. we can immediately deduce the existence of extremals for
anisotropic Trudinger-Moser inequality on bounded domain. Finally, we also
consider the optimal concentration level of the anisotropic singular
Trudinger-Moser functional. The method is based on the limiting Hardy-Sobolev
approximation method and constructing a suitable normalized anisotropic
concentrating sequence
Existence of the normalized solutions to the nonlocal elliptic system with partial confinement
Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
Let II be a bounded interval of R{\mathbb{R}} and λ1(I){\lambda }_{1}\left(I) denote the first eigenvalue of the nonlocal operator (−Δ)14{(-\Delta )}^{\tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0⩽α<λ1(I)0\leqslant \alpha \lt {\lambda }_{1}(I), there holds supu∈W012,2(I),‖(−Δ)14u‖22−α∥u∥22≤1∫Ieπu2dx<+∞,\mathop{\sup }\limits_{u\in {W}_{0}^{\frac{1}{2},2}(I),\Vert {\left(-\Delta )}^{\tfrac{1}{4}}u{\Vert }_{2}^{2}-\alpha {\parallel u\parallel }_{2}^{2}\le 1}\mathop{\int }\limits_{I}{e}^{\pi {u}^{2}}{\rm{d}}x\lt +\infty , and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions
Multiple Nontrivial Solutions for a Class of Biharmonic Elliptic Equations with Sobolev Critical Exponent
In this paper, we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic equation with Sobolev critical exponent in a bounded domain. By using the idea of the previous paper, we generalize the results and prove the existence and multiplicity of nontrivial solutions of the biharmonic elliptic equations
Orlicz Regularity for Non-Divergence Parabolic Systems with Partially Vmo Coefficients
This work treats the interior Orlicz regularity for strong solutions of a class of non-divergence parabolic systems with coefficients just measurable in time and VMO in the spatial variables