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The paper is devoted to the question as to how "bad" the junior coefficients of elliptic and parabolic equations may be in order that classical properties of their solutions (such as the strict maximum principle, the Harnack inequality and the Liouville theorem) still occur. The answers are given in terms of the Lebesgue and Morrey spaces.2010 Mathematics Subject Classification. Primary 35B50, 35B53, 35B45.

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface.The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the "mean curvature" condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex "in the mean") then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of JenkinsSemn. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however. existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time f 2 0. In addition, we establish estimates of the rate at which solutions tend to zero as t -+ 03.

Various classes of non-uniformly elliptic (and parabolic) equations of second order of the form ) we have shown that this method is applicable to the whole class of uniformly elliptic and parabolic equations. In the present paper we investigate the possibility of applying it to non-uniformly elliptic and parabolic equations. I t turns out that it is applicable, roughly speaking, to those classes of [I] for which the order of nonuniformity of the quadratic form aij(x, u, is less than two. The first part of this paper is devoted to the proof of this assertion.In the second part we analyze a different method of obtaining local estimates for 1u. I which is applicable to elliptic equations of the form in terms of m a n IuI and embraces such interesting cases as equations for the mean curvature of a 1 We shall use the notation 677