231 research outputs found
Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth
We establish global regularity for weak solutions to quasilinear divergence
form elliptic and parabolic equations over Lipschitz domains with controlled
growth conditions on low order terms. The leading coefficients belong to the
class of BMO functions with small mean oscillations with respect to .Comment: 24 pages, to be submitte
L^p and Schauder estimates for nonvariational operators structured on H\"ormander vector fields with drift
We consider linear second order nonvariational partial differential operators
of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the
X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying
H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded
measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly
positive. We prove that if the coefficients a_{ij} are H\"older continuous with
respect to the distance induced by the vector fields, then local Schauder
estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space
VMO with respect to the distance induced by the vector fields, then local L^{p}
estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the
presence of the drift term X_{0}, so that our class of operators covers, for
instance, Kolmogorov-Fokker-Planck operators
Attuale formulazione della teoria degli operatori vicini e attuale definizione di operatore ellittico
See directly the article
Partial regularity for the Navier-Stokes-Fourier system
This paper addresses a nonstationary flow of heat-conductive incompressible
Newtonian fluid with temperature-dependent viscosity coupled with linear heat
transfer with advection and a viscous heat source term, under Navier/Dirichlet
boundary conditions. The partial regularity for the velocity of the fluid is
proved to each proper weak solution, that is, for such weak solutions which
satisfy some local energy estimates in a similar way to the suitable weak
solutions of the Navier-Stokes system. Finally, we study the nature of the set
of points in space and time upon which proper weak solutions could be singular.Comment: 25 pages, v2: Navier/Dirichlet boundary conditions replace
homogeneous Dirichlet boundary condition
Holder continuity for a drift-diffusion equation with pressure
We address the persistence of H\"older continuity for weak solutions of the
linear drift-diffusion equation with nonlocal pressure u_t + b \cdot \grad u
- \lap u = \grad p,\qquad \grad\cdot u =0 on ,
with . The drift velocity is assumed to be at the critical
regularity level, with respect to the natural scaling of the equations. The
proof draws on Campanato's characterization of H\"older spaces, and uses a
maximum-principle-type argument by which we control the growth in time of
certain local averages of . We provide an estimate that does not depend on
any local smallness condition on the vector field , but only on scale
invariant quantities
Partial Schauder estimates for second-order elliptic and parabolic equations
We establish Schauder estimates for both divergence and non-divergence form
second-order elliptic and parabolic equations involving H\"older semi-norms not
with respect to all, but only with respect to some of the independent
variables.Comment: CVPDE, accepted (2010)
Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
The main result of this paper concerns the behavior of a free boundary
arising from a minimization problem, close to the fixed boundary in two
dimensions
Riesz potentials and nonlinear parabolic equations
The spatial gradient of solutions to nonlinear degenerate parabolic equations
can be pointwise estimated by the caloric Riesz potential of the right hand
side datum, exactly as in the case of the heat equation. Heat kernels type
estimates persist in the nonlinear cas
Hardy spaces of the conjugate Beltrami equation
We study Hardy spaces of solutions to the conjugate Beltrami equation with
Lipschitz coefficient on Dini-smooth simply connected planar domains, in the
range of exponents . We analyse their boundary behaviour and certain
density properties of their traces. We derive on the way an analog of the Fatou
theorem for the Dirichlet and Neumann problems associated with the equation
with -boundary data
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