From Non-symmetric Particle Systems to Non-linear PDEs on Fractals

Chen, Joe P.; Hinz, Michael; Teplyaev, Alexander

doi:10.1007/978-3-319-74929-7_34

“…Several tools used in the present paper rely heavily on the use of linear and harmonic functions, and second order versions are not so straightforward to see. A question in a different direction, particularly interesting in connection with probability [16], is how to approximate equations involving nonlinear first order terms. There are results on the convergence of certain non-linear operators along varying spaces [98], but they do not cover these cases.…”

Section: Short Remarks On Possible Generalizationsmentioning

confidence: 99%

Approximation of partial differential equations on compact resistance spaces

Hinz

¹

,

Meinert

²

2021

Calc. Var.

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We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

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“…Several tools used in the present paper rely heavily on the use of linear and harmonic functions, and second order version are not so straightforward to see. A fourth natural question to ask, in particular in connection with related problems in probability, [16], is how to approximate equations involving nonlinear first order terms. Although there are results on the convergence of certain non-linear operators along varying spaces, [98], they do not cover these cases.…”

Section: 5mentioning

confidence: 99%

Approximation of partial differential equations on compact resistance spaces

Hinz¹,

Meinert²

2020

Preprint

Self Cite

0

View full text Add to dashboard Cite

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

show abstract