相关论文: On the extensions of the De Giorgi approach to non…

In this paper we discuss an extension of some results obtained by E. Serra and P. Tilli, in [Serra&Tilli '12, Serra&Tilli '16], concerning an original conjecture by E. De Giorgi ([De Giorgi '96, De Giorgi '06]) on a purely minimization…

We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a…

We deal with the De Giorgi H{\"o}lder regularity theory for parabolic equations with rough coefficients and parabolic De Giorgi classes which extend the notion of solution. We give a quantitative proof of the interior H{\"o}lder regularity…

In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the…

The theory of De Giorgi (1958) and Nash (1959) solves Hilbert's 19th problem and constitutes a major advance in the analysis of PDEs in the 20th century. This theory concerns the H\"older regularity of solutions to elliptic and parabolic…

This book presents a comprehensive regularity theory for solutions of elliptic, parabolic, and kinetic equations. The foundation of this theory was laid by E. De Giorgi's groundbreaking resolution of Hilbert's nineteenth problem in 1956.…

We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…

In this note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.

In this paper, we establish the existence of solutions for a particular class of degenerate hyperbolic equations. Following this, we approximate these degenerate equations by employing a sequence of uniformly hyperbolic equations. Notably,…

An $hp$-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, {\it a priori} $hp$-error estimates…

H\"older estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.

We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…

We consider several classes of degenerate hyperbolic equations involving delay terms and suitable nonlinearities. The idea is to rewrite the problems in an abstract way and, using semigroup theory and energy method, we study well posedness…

Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic equations. As by-product of these estimates, we get an H\"older estimate for bounded…

For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.

This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. Throughout…

By borrowing ideas from the parabolic theory, we use a combination of De Giorgi's and Moser's methods to give some remarks on the proof of H\"older continuity of weak solutions of elliptic equations.

In this paper we develop a systematic reduction procedure for determining intermediate integrals of second order hyperbolic equations so that exact solutions of the second order PDEs under interest can be obtained by solving first order…

In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a $hp$-version discontinuous Galerkin finite element approximation in the time…

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic…