Parabolic oblique derivative problem in generalized Morrey spaces

We study the regularity of the solutions of the oblique derivative problem for linear uniformly parabolic equations with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized Morrey space than the strong solution belongs to the corresponding generalized Sobolev-Morrey space

On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.Comment: 44 pages, introduction revised, references expanded. To appear in Arch. Rational Mech. Ana

Singularities of Nonlinear Elliptic Systems

Through Morrey's spaces (plus Zorko's spaces) and their potentials/capacities as well as Hausdorff contents/dimensions, this paper estimates the singular sets of nonlinear elliptic systems of the even-ordered Meyers-Elcrat type and a class of quadratic functionals inducing harmonic maps.Comment: 18 pages Communications in Partial Differential Equation

A note on boundedness of operators in Grand Grand Morrey spaces

In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calder\'on-Zygmund operators in the framework of grand grand Morrey spaces.Comment: 8 page

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)

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 $x$.Comment: 24 pages, to be submitte

SF3B1-mutated chronic lymphocytic leukemia shows evidence of NOTCH1 pathway activation including CD20 downregulation

Chronic lymphocytic leukemia (CLL) is characterized by a low CD20 expression, in part explained by an epigenetic-driven downregulation triggered by mutations of the NOTCH1 gene. In the present study, by taking advantage of a wide and well-characterized CLL cohort (n=537), we demonstrate that CD20 expression is downregulated in SF3B1-mutated CLL in an extent similar to NOTCH1-mutated CLL. In fact, SF3B1-mutated CLL cells show common features with NOTCH1-mutated CLL cells, including a gene expression profile enriched of NOTCH1-related gene sets and elevated expression of the active intracytoplasmic NOTCH1. Activation of the NOTCH1 signaling and down-regulation of surface CD20 in SF3B1-mutated CLL cells correlate with over-expression of an alternatively spliced form of DVL2, a component of the Wnt pathway and negative regulator of the NOTCH1 pathway. These findings are confirmed by separately analyzing the CD20-dim and CD20-bright cell fractions from SF3B1-mutated cases as well as by DVL2 knock-out experiments in CLL-like cell models. Altogether, the clinical and biological features that characterize NOTCH1-mutated CLL may also be recapitulated in SF3B1-mutated CLL, contributing to explain the poor prognosis of this CLL subset and providing the rationale for expanding novel agents-based therapies to SF3B1-mutated CLL

A limit model for thermoelectric equations

We analyze the asymptotic behavior corresponding to the arbitrary high conductivity of the heat in the thermoelectric devices. This work deals with a steady-state multidimensional thermistor problem, considering the Joule effect and both spatial and temperature dependent transport coefficients under some real boundary conditions in accordance with the Seebeck-Peltier-Thomson cross-effects. Our first purpose is that the existence of a weak solution holds true under minimal assumptions on the data, as in particular nonsmooth domains. Two existence results are studied under different assumptions on the electrical conductivity. Their proofs are based on a fixed point argument, compactness methods, and existence and regularity theory for elliptic scalar equations. The second purpose is to show the existence of a limit model illustrating the asymptotic situation.Comment: 20 page