Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed 𝓐-p quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone of 𝓐.

Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertexOof the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance toO. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

ON THE CONE OF CURVES AND OF LINE BUNDLES OF A RATIONAL SURFACE

Let Z be a smooth projective rational surface. A condition that implies the polyhedrality of the cone of curves of Z is given. This one depends only on the configuration of infinitely near points associated with the morphism which provides Z from a relatively minimal model X and it holds for a wide range of surfaces whose anticanonical bundle is not ample. When the above configuration is a chain, the condition consists uniquely on deciding whether certain datum is positive. Furthermore, we study polyhedrality and regularity of the characteristic cone and of the cone of curves of Z for some particular cases.

New singular solutions of Protter's problem for the3D wave equation

In 1952, for the wave equation,Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a3D domainΩ0,bounded by two characteristic conesΣ1andΣ2,0and a plane regionΣ0. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right-hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet's or Neumann's conditions onΣ0. In the present paper, we consider the case of third BVP onΣ0and obtain the existence of many singular solutions for the wave equation. Especially, for Protter's problems inℝ3, it is shown here that for anyn∈ℕthere exists aCn(Ω¯0)- right-hand side function, for which the corresponding unique generalized solution belongs toCn(Ω¯0\O),but has a strong power-type singularity of ordernat the pointO. This singularity is isolated only at the vertexOof the characteristic coneΣ2,0and does not propagate along the cone.