# Partial differential equations and systems related to Morrey spaces

Sammanfattning: This PhD thesis deals with the study of well posedness, existence and regularity properties of solutions of partial differential equations and systems. Preparatory to the study of partial differential equations is the action of some integral operators, that are extensively used. Such results are very useful to obtain regularity properties of solutions of elliptic, parabolic and ultraparabolic equations of second order with discontinuous coefficients, and later of systems. The thesis consists of five papers (Paper A -- Paper F), an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics, and an appendix, where the behavior of Morrey spaces in connection with a lot of other spaces is presented and discussed. In paper A we study the local regularity in the Lebesgue spaces $L^{p},$ $1systems of arbitrary order in nondivergence form with coefficients, which can be discontinuous. In the case of continuous coefficients we review, discuss and complement the results obtained by S. Agmon, A. Douglis and L. Nirenberg in 1959, 1964 and by S. Campanato in 1977. In paper B we study the Cauchy-Dirichlet problem related to a linear parabolic equation of second order in divergence form with discontinuous coefficients. Moreover, we prove estimates in the space $\,\,H^{1- \frac1p},\,\,$ for every $\,1\,<\,p\,<\,\infty.\,$ The $H^{\frac12}$ local regularity of $u$ was also studied by A. Marino and A. Maugeri in 1989. Moreover, they obtained relevant results about $L^{p}-$local regularity for the gradient of $u$ and existence of solutions of a nonhomogeneous Cauchy-Dirichlet problem. We emphasize that the regularity we study in this paper is not of the type considered by N. Meyers in 1963 and by M. Giaquinta in 1983. These papers are concerned only with $p$ close to $2.$ Our results are in fact true for any value of $p$ in the range $]1,\,+ \infty[.$ Paper C is devoted to the study of a Cauchy-Dirichlet problem related to nondivergence form parabolic equation with discontinuous coefficients and more precisely we derive and discuss existence, uniqueness and regularity of the solution. This problem is inspired by the study made by M. Bramanti and C. Cerutti in 1993. We point out that our fundamental tools are some properties related to the products of the lower order terms with the solution $u $ and its derivatives. The technique used to derive these results is based on dividing a cylinder in sections and obtaining the requested estimates in each part of the subdivision. In paper D we derive and discuss some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals. The considered kind of functionals arise as the energy of maps between Riemannian manifolds. From this point of view, the geometric interest may occur on the above functionals. Moreover, we observe that some methods of proofs of regularity for classes of nonlinear elliptic systems can also be applied to a lot of equations in nonlinear Hodge theory. Paper E deals with the study of some qualitative properties of positive solutions of a chr\"odinger type equation of the kind $ L u+V u=0, $ having discontinuous coefficients, where $L $ is the Kolmogorov operator in $\R^{n+1} $ and $V$ belongs to a Stummel-Kato class. Moreover, we consider the Green function for the constant coefficients operator $L_0 $ and build the Green function for the operator $L. $ We also produce the proof of interior regularity and a uniqueness result for the Cauchy-Dirichlet problem associated to $L, $ making in both cases the additional assumption that the solution $u$ is bounded. Moreover, a density argument allows us to remove the extra condition of boundedness on $u$ from the uniqueness result and then we use this fact to remove the additional assumption from the regularity theorem. To motivate our interest in this kind of operators, we recall that they arise e.g. in the stochastic theory and in the theory of diffusion processes. For instance, the linear Fokker-Planck equation can be written as a particular case of the above equation.

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