1,720,993 research outputs found
The Wolff gradient bound for degenerate parabolic equations
No full textThe spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of p-Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure
A nonlinear Stein theorem
No full textFor vector valued solutions u to the p-Laplacian system −△pu=F in a domain of Rn,p>1,n≥2, if F belongs to the limiting Lorentz space L(n,1), then Du is continuous
Pointwise gradient estimates
No full textWe survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear, possibly degenerate, elliptic and parabolic equation
Universal potential estimates
No full textAbstractWe prove a class of endpoint pointwise estimates for solutions to quasilinear, possibly degenerate elliptic equations in terms of linear and nonlinear potentials of Wolff type of the source term. Such estimates allow to bound size and oscillations of solutions and their gradients pointwise, and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients. In particular, local estimates in Hölder, Lipschitz, Morrey and fractional spaces, as well as Calderón–Zygmund estimates, follow as a corollary in a unified way. Moreover, estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied. The classical Wolff potential estimate by Kilpeläinen & Malý and Trudinger & Wang as well as recent Wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases
New perturbation methods for nonlinear parabolic problems
No full textAbstractWe develop methods aimed at deriving regularity results for solutions to nonlinear degenerate parabolic equations and systems via local perturbation; as a consequence we obtain, in a unified way, Lipschitz continuity of solutions under weak parabolicity assumptions, and gradient continuity results in borderline cases. Nonlinear Schauder estimates as those of Misawa (2002) [29] are recovered and extended to more general settings
Gradient regularity for nonlinear parabolic equations
Full text linkWe consider non-homogeneous degenerate and singular parabolic equations of the p-Laplacian type and prove pointwise bounds for the spatial gradient of solutions in terms of intrinsic parabolic potentials of the given datum. In particular, the main estimate found reproduces in a sharp way the behavior of the Barenblatt (fundamental) solution when applied to the basic model case of the evolutionary p-Laplacian equation with Dirac datum. Using these results as a starting point, we then give sufficient conditions to ensure that the gradient is continuous in terms of potentials; in turn these imply borderline cases of known parabolic results and the validity of well-known elliptic results whose extension to the parabolic case remained an open issue. As an intermediate result we prove the Ho ̈lder continuity of the gradient of solutions to possibly degenerate, homo- geneous and quasilinear parabolic equations defined by general operators
Guide to nonlinear potential estimates
No full textOne of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measure data problems
- …
