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In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $div(A(x)\nabla u(x))=0$, where the symmetric matrix $A(x)$ satisfies the condition ${|\xi |}^2+{|A(x)\xi |}^2\le K(x)⟨A(x)\xi ,\xi ⟩$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp (\beta K(x))$ is integrable for some $\beta >0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L_{\text{loc}}^2{\log }^{\alpha -1}L$, . Moreover we show exactly how the degree of improved regularity depends on $\beta$.

We introduce a sort of "local" Morrey spaces and show an existence and uniqueness theorem for the Dirichlet problem in unbounded domains for linear second order elliptic partial differential equations with principal coefficients "close" to functions having derivatives in such spaces.

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by $loglog{(1/|x\left|\right)}^{-1}$. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann....