Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On the Correctness of the Dirichlet Problem for Equations with Hessian Operators

Author(s):

Svetlana Ivanovna Prokof'eva

Ph.D., Associate Professor of the Mathematics Department of the St. Petersburg State University of Architecture and Civil Engineering (SPbGASU).

prokof_1960@mail.ru

Galina Vladimirovna Yakunina

Ph.D., Associate Professor of the Mathematics Department of the St. Petersburg State University of Architecture and Civil Engineering (SPbGASU).

yakuninagalina@yandex.ru

Tatiana Vladimirovna Ryabikova

Ph.D., Head of the Mathematics Department of the St. Petersburg State University of Architecture and Civil Engineering (SPbGASU).

tanya.dovid@gmail.com

Abstract:

The article deals with completely nonlinear partial differential equations of the second order, called Hessian, i.e. depending only on the second derivative solutions. The importance of the concept of ellipticity for linear and Hessian differential equations is compared. The work gives an example of an equation showing that the ellipticity of the operator is not correct for the solvability of the Dirichlet problem. An alternative set is proposed – a cone of positive monotonicity of the operator, on which the Dirichlet problem has a unique solution. It follows from the example that this equation has at least two different solutions satisfying the ellipticity condition, but one of them is from the cone and the other is not included in the cone.

Keywords

• ellipticity
• Garding cone
• Hessian equations
• monotone operator

References:

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