We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new dualities between preordered structures and locales or topological spaces can be generated through our topos-theoretic machinery in a uniform way. We then apply our topos-theoretic interpretation to obtain results connecting properties of preorders and properties of the corresponding locales or topological spaces, and we establish adjunctions between various kinds of categories as natural applications of our general methodology. In the last part of the paper, we exploit the theory developed in the previous parts to obtain a topos-theoretic interpetation of the problem of finding explicit descriptions of models of 'ordered algebraic theories' presented by generators and relations, and give several examples which illustrate the effectiveness of our methodology. In passing, we provide a number of other applications of our theory to Algebra, Topology and Logic.

The note concerns on some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals of the form

\(\int_{\Omega}F(x,u,Du)dx\)

where the integrand has the following special form

\(F(x,u,Du)=A(x,u,g^{\alpha\beta}(x)h_{ij}(u)\frac{\partial{u}^i}{\partial{x}^\alpha}\frac{\partial{u}^i}{\partial{x}^\beta})\)

where \(g^{\alpha\beta})\) and \((h_{ij})\) symmetric positive denite matrices. We are not assuming the continuity of \(A\) and \(g\) with respect to \(x\). We suppose that \(A(\cdot , u,t)/(1+t)\) and \(g(\cdot)\) are in the class \(L^\infty \cap VMO\).