Abstract We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton–Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $$\partial _t u+H(x,t,Du,D^2u)=0$$ ∂ t u + H ( x , t , D u , D 2 u ) = 0 in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $$\varepsilon F(x,t,D^2u)$$ ε F ( x , t , D 2 u ) . Assuming that $$u\in C^\alpha _x$$ u ∈ C x α , $$u_0\in C^\eta $$ u 0 ∈ C η , $$H\in C^\beta _x$$ H ∈ C x β and having power growth $$\gamma $$ γ in the gradient entry, we establish a convergence rate of order $$\varepsilon ^{\min \left\{ \frac{\eta }{2},\frac{\beta +\gamma (\alpha -1)}{\beta +\gamma (\alpha -1)+2-\alpha }\right\} }$$ ε min η 2 , β + γ ( α - 1 ) β + γ ( α - 1 ) + 2 - α . Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on H . The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with Hölder coefficients.

Abstract In this paper we study the geometry of generalized $$\varphi $$ φ -vacuum static spaces, proving estimates for the $$\varphi $$ φ -scalar curvature and for the first eigenvalue of the Jacobi operator, and also rigidity under various geometric ass be a compact manifold. umptions; in particular, we prove a result related to the famous Cosmic no-hair conjecture of Boucher, Gibbons and Horowitz.

Abstract In any dimension $$n+1\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> we construct a sequence of closed $$(n+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional Riemannian manifolds with positive Ricci curvature admitting embedded two-sided minimal hypersurfaces such that the following hold: (i) any such hypersurface has Morse index one; (ii) the first Betti numbers of the hypsersurfaces are not uniformly bounded along the sequence.

Abstract We investigate polyconvexity of the double well function $$f(X):= |X-X_1|^2|X-X_2|^2$$ f ( X ) : = | X - X 1 | 2 | X - X 2 | 2 for given matrices $$X_1, X_2 \in \mathbb {R}^{n \times n}$$ X 1 , X 2 ∈ R n × n . Such functions are fundamental in the modeling of phase transitions in materials, but their non-convex nature presents challenges for the analysis of variational problems. Polyconvexity of f is related to the singular values of the matrix difference $$X_1 - X_2$$ X 1 - X 2 . We prove that f is polyconvex if and only if the square of the largest singular value does not exceed the sum of the squares of the other singular values. This condition allows the function to be decomposed into the sum of a strictly convex part and a null Lagrangean. As a direct application of this result, we prove an existence and uniqueness theorem for the corresponding Dirichlet minimization problem of the integral functional.

Abstract We establish forward, backward and elliptic Harnack inequalities for non-negative solutions to a class of doubly non-linear parabolic partial differential equations. These Harnack estimates are established in a proper range of parameters p and q below. Such a range is shown to be optimal for a Harnack estimate to hold. Quantitative boundedness estimates for solutions and an expansion of positivity result for non-negative super-solutions are instrumental in the proof.