1,147 research outputs found
Sharp Regularity for Weak Solutions to the Porous Medium Equation
Let be a nonnegative, local, weak solution to the porous medium equation
for in a space-time cylinder . Fix a point
: if the average
a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0,
then the quantity is locally bounded in a proper
cylinder, whose center lies at time . This implies that in the
same cylinder the solution is H\"older continuous with exponent
, which is known to be optimal. Moreover, presents a
sort of instantaneous regularisation, which we quantify
Self-improving property of the fast diffusion equation
We show that the gradient of the -power of a solution to a singular
parabolic equation of porous medium-type (also known as fast diffusion
equation), satisfies a reverse H\"older inequality in suitable intrinsic
cylinders. Relying on an intrinsic Calder\'on-Zygmund covering argument, we are
able to prove the local higher integrability of such a gradient for
. Our estimates are satisfied for a
general class of growth assumptions on the non linearity. In this way, we
extend the theory for (see [GS16] in the list of references) to the
singular case. In particular, an intrinsic metric that depends on the solution
itself is introduced for the singular regime.Comment: arXiv admin note: text overlap with arXiv:1603.0724
A Boundary Estimate for Singular Parabolic Diffusion Equations
We prove an estimate on the modulus of continuity at a boundary point of a
cylindrical domain for local weak solutions to singular parabolic equations of
p-laplacian type. The estimate is given in terms of a Wiener-type integral,
defined by a proper elliptic p-capacity
A Boundary Estimate for Degenerate Parabolic Diffusion Equations
We prove an estimate on the modulus of continuity at a boundary point of a
cylindrical domain for local weak solutions to degenerate parabolic equations
of -laplacian type. The estimate is given in terms of a Wiener-type
integral, defined by a proper elliptic -capacity.Comment: arXiv admin note: text overlap with arXiv:1703.0490
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