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tfg_Lleal_Sirvent_Adria.pdf (459.34 KB)

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Bachelor thesis

Publication date

2024-06-10

Publication license

cc-by-nc-nd (c) Adrià Lleal Sirvent, 2024
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/220153

Poisson’s equation and eigenfunctions of the Laplacian

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Abstract

This work aims to explore the foundations of partial differential equations (PDEs) by focusing specifically on Poisson’s equation with Dirichlet boundary conditions and the eigenvalue problem for the Laplacian. These equations are of special interest in both mathematics and physics. Although they are among the simplest cases of PDEs, they introduce techniques and results that are key to solving more complex equations. In particular, we will introduce the weak formulation of both equations and prove the existence of weak solutions in two different ways. The first method uses Hilbert space techniques, such as the Lax-Milgram theorem and the Spectral theorem, while the second method involves the minimization of functionals. Ultimately, we will study the regularity of weak solutions and examine a practical case in which the previous theory is very useful.

Description

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2024, Director: Tomás Sanz Perela

Subject

Equacions en derivades parcials, Laplacià, Problema de Dirichlet, Treballs de fi de grau

Subject (English)

Partial differential equations, Laplacian operator, Dirichlet problem, Bachelor's theses

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Treballs Finals de Grau (TFG) - Matemàtiques

Citation

LLEAL SIRVENT, Adrià. Poisson’s equation and eigenfunctions of the Laplacian. [consulted: 19 of August of 2026]. Available at: https://hdl.handle.net/2445/220153

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