2018/02/26 by Karthik Adimurthi, Adimurthi, Karthik, Sun‐Sig Byun +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1802.09175
We prove boundary higher integrability for the (spatial) gradient of\n\very weak solutions of quasilinear parabolic equations of the form \n
left
\n \ut - div
mathcalA(x,t,
nabla u) = 0 · amp;
quad
texton
\n
Omega
times (-T,T),
u = 0 · amp;
quad
texton
partial
Omega
times\n(-T,T),\n \
right. where the non-linear structure \A(x,\nt,\∇ u) is modelled after the variable exponent p(x,t)-Laplace operator\ngiven by |\∇ u|p(x,t)-2 \∇ u. To this end, we prove that the\ngradients satisfy a reverse H "older inequality near the boundary by\nconstructing a suitable test function which is Lipschitz continuous and\npreserves the boundary values. In the interior case, such a result was proved\nin citebogelein2014very provided p(x,t) \≥ mathfrakp- \≥ 2 holds\nand was then extended to the singular case \(2n)/(n+2)< mathfrakp-\≤\np(x,t)\≤ mathfrakp+ \≤ 2 in citeli2017very. This restriction was\nnecessary because the intrinsic scalings for quasilinear parabolic problems are\ndifferent in the case mathfrakp+ \≤ 2 and mathfrakp-\≥ 2.\n In this paper, we develop a new unified intrinsic scaling, using which, we\nare able to extend the results of citebogelein2014very,li2017very to the\nfull range \(2n)/(n+2) < mathfrakp- \≤ p(x,t)\≤\n mathfrakp+<\∞ and also obtain analogous results upto the boundary.\n\The main novelty of this paper is that our methods are able to handle\nboth the singular case and degenerate case simultaneously. To simplify the\nexposition, we will only prove the higher integrability result near the\nboundary, provided the domain \Ω satisfies a uniform measure density\ncondition. Our techniques are also applicable to higher order equations as well\nas systems.\n