Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation

• Autorzy: Penent Guilaume , Privault Nicolas
• Języki publikacji: EN

Abstrakty

EN

We provide sufficient conditions for the existence of classical solutions of fractional semilinear elliptic PDEs of index α ∈ (1,2) with polynomial gradient nonlinearities on d-dimensional balls, d ≥ 2. Our approach uses a tree-based probabilistic representation of solutions and their partial derivatives using α-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. In comparison with the existing literature on the regularity of solutions, no polynomial order condition is imposed on gradient nonlinearities. Numerical illustrations demonstrate the accuracy of the method in dimension d=10, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.

Słowa kluczowe

EN

elliptic PDEs; semilinear PDEs; fractional Laplacian; gradient nonlinearities; stable processes; branching processes; Monte-Carlo method

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2024
• Tom: Vol. 44, Fasc. 2
• Strony: 211--235
• Opis fizyczny: Bibliogr. 33 poz., wykr.

Twórcy

autor

Penent Guilaume

• Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371

autor

Privault Nicolas

• Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371

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