We’d like to recall that a common feature for nonlinear problems with growth rate \(p>2\) is that the higher differentiability is proven for a nonlinear expression of the gradient which takes into account the growth of the principal part of the equation. ![]() Indeed, the higher integrability of the spatial gradient of weak solutions to equation ( 1.1) has been proven in, under suitable assumptions on the datum f in the scale of Sobolev spaces. This strategy has revealed to be successful also for widely degenerate equations as in ( 1.1). To make it more clear what a functional is, we compare it to functions. In a very short way, a functional is a function of a function. So in order to understand the method of calculus of variations, we rst need to know what functionals are. It is worth mentioning that the higher integrability of the gradient of the solution is achieved through an interpolation argument, once its higher differentiability is established. Calculus of variations is a subject that deals with functionals. In the above mentioned papers, the problem has been faced or in case of homogeneous equations or considering sufficiently regular datum. in in case of superquadratic growth, while Scheven in faced the subquadratic growth case. These questions have been exploited in case of non-degenerate parabolic problems with quadratic growth by Campanato in, by Duzaar et al. a function whose argument is another function (or more functions) and whose outcome is a uniquely assigned number. More difficult to grasp is the idea of a function of functions, i.e. In this paper we address two interrelated aspects of the regularity theory for solutions to parabolic problems, namely the higher differentiability and the higher integrability of the gradient of the weak solutions to ( 1.1), with the main aim of weakening the assumption on the datum f with respect to the available literature. 1.4 The Calculus of Variations How do we nd the function y(x) which minimises, or more generally makes stationary, our archetypal functional Fy Z b a f(x,y,y0)dx, with xed values of y at the end-points (viz. Calculus of Variations (I) 1 Introduction The generic concept of function of one or several variables is an important and well established Calculus notion. ![]() The main feature of this equation is that it possesses a wide degeneracy, coming from the fact that its modulus of ellipticity vanishes at all points where \(\left| Du\right| \le 1\) and hence its principal part behaves like a p-Laplacian operator only for large values of \(\left| Du\right| \). The func-tion f creates a one-to-one correspondencebetween these two sets, denoted as y. In the analysis of functions the focus is on the relation between two sets of numbers, the independent (x) and the dependent (y) set. For a precise description of this motivation we refer to and. The foundations of calculus of variations The problem of the calculus of variations evolves from the analysis of func-tions. Which appears in gas filtration problems taking into account the initial pressure gradient.
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