Professional paper

Weak solutions of partial differential equations

Krešimir Burazin orcid.org/0000-0001-6713-7560 ; Fakultet primijenjene matematike i informatike, Sveučilište J. J. Strossmayera u Osijeku *
Maja Damjanović orcid.org/0000-0002-1964-5672 ; Fakultet primijenjene matematike i informatike, Sveučilište J. J. Strossmayera u Osijeku

* Corresponding author.

Abstract

In this paper, we consider the concept of a weak solution for an elliptic partial differential equation that models various phenomena in nature, such as steady-state heat conduction, and the equilibrium of elastic membranes. First, we discuss the need for introducing a generalized concept of a solution, and then introduce the concept of weak derivatives and Sobolev spaces, which allow for the rigorous definition of a weak solution. Using the Lax-Milgram theorem, we demonstrate the existence and uniqueness of the weak solution, as well as the equivalence of the variational formulation with the minimization of the
energy functional.

Keywords

weak derivative, weak solution, boundary value problem, Sobolev spaces, variational equation, energy functional

Hrčak ID:

340755

URI

https://hrcak.srce.hr/340755

Publication date:

10.12.2025.

Article data in other languages: croatian

Visits: 509 *