Browsing by Author "Elenius, Theo"

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    Algebraic Geometry of Deep Polynomial Neural Networks
    (2022-09-02) Elenius, Theo; Kubjas, Kaie; Perustieteiden korkeakoulu; Kubjas, Kaie
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    Regularity for nonhomogeneous variational problems on metric measure spaces
    (2023-12-12) Elenius, Theo; Kinnunen, Juha; Perustieteiden korkeakoulu; Kinnunen, Juha
    This thesis studies regularity for nonhomogeneous variational problems in the setting of metric measure spaces. Assuming that the space satisfies a Poincaré inequality and that the measure is doubling, quasi-minimizers of variational integrals satisfy additional regularity properties than assumed a priori. In particular, the quasi-minimizers are locally Hölder continuous and satisfy Harnack’s inequality. Following De Giorgi’s method, the essential tools in proving the regularity properties are energy estimates for the quasi-minimizers. The model used in this thesis is a slight generalization of what exists in the current literature and the thesis simultaneously presents a detailed account of the proofs and keeps track of the dependencies of constants.