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1.

Avelin, Benny

et al.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We prove a comparison principle for the porous medium equation in more general open sets in Rn+1 than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

We prove local Calderon-Zygmund estimates for weak solutions of the evolutionary p(x, t)-Laplacian system partial derivative(t)u - div (a(x, t)vertical bar Du vertical bar(p(x,t)-2)Du) = div (vertical bar F vertical bar F-p(x,F-t)-2 under the classical hypothesis of logarithmic continuity for the variable exponent p(x, t). More precisely, we show that the spatial gradient Du of the solution is as integrable as the right-hand side F, i.e., vertical bar F vertical bar(p(.)) is an element of L-loc(q) double right arrow vertical bar Du vertical bar(p(.)) is an element of L-loc(q) for any q > 1, together with quantitative estimates. Thereby we allow the presence of eventually discontinuous coefficients a(x, t), requiring only a VMO condition with respect to the spatial variable x.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

On a parabolic symmetry problem2007In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 23, p. 513-536Article in journal (Refereed)

4.

Nyström, Kaj

et al.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Strömqvist, Martin

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We prove the existence of big pieces of regular parabolic Lipschitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized in time two cube condition. An application to the fine properties of parabolic measure is given.

We prove the existence of big pieces of regular parabolic Lip-schitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized two cube condition. An application to the fine properties of parabolic measure is given.