2018/04/10 by Matteo Bonforte, Bonforte, Matteo, Nikita Simonov +1
Computer Science · Mathematics · #35B45 #35B65 #35K55 #35K65 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1804.03537
openalex publication_date 2018/04/10 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We study a priori estimates for a class of non-negative local weak solution\nto the weighted fast diffusion equation ut = |x|\γ \∇\⋅\n(|x|-\β \∇ um), with 0 < m <1 posed on cylinders of\n(0,T)\× mathbb RN. The weights |x|\γ and |x|-\β, with\n\γ < N and \γ -2 < \β \≤ \γ(N-2)/N can be both degenerate\nand singular and need not belong to the class \A2, a typical\nassumption for this kind of problems. This range of parameters is optimal for\nthe validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play\nthe role of the standard Sobolev inequalities in this more complicated weighted\nsetting.\n The weights that we consider are not translation invariant and this causes a\nnumber of extra difficulties and a variety of scenarios: for instance, the\nscaling properties of the equation change when considering the problem around\nthe origin or far from it. We therefore prove quantitative - with computable\nconstants - upper and lower estimates for local weak solutions, focussing our\nattention where a change of geometry appears. Such estimates fairly combine\ninto forms of Harnack inequalities of forward, backward and elliptic type. As a\nconsequence, we obtain H "older continuity of the solutions, with a\nquantitative (even if non-optimal) exponent. Our results apply to a quite large\nvariety of solutions and problems. The proof of the positivity estimates\nrequires a new method and represents the main technical novelty of this paper.\n Our techniques are flexible and can be adapted to more general settings, for\ninstance to a wider class of weights or to similar problems posed on Riemannian\nmanifolds, possibly with unbounded curvature. In the linear case, m=1, we\nalso prove quantitative estimates, recovering known results in some cases and\nextending such results to a wider class of weights.\n