Factorization of linear and nonlinear differential operators: necessary and sufficient conditions [PDF]
arXiv, 2011 An algebraic approach for factorizing nonlinear partial differential equations (PDEs) and systems of PDEs is provided. In the particular case of second order linear and nonlinear PDEs and systems of PDEs, necessary and sufficient conditions of factorization are given.
arxiv
Existence of $C^α$ solutions to integro-PDEs [PDF]
arXiv, 2017 This paper is concerned with existence of a $C^{\alpha}$ viscosity solution of a second order non-translation invariant integro-PDE. We first obtain a weak Harnack inequality for such integro-PDE. We then use the weak Harnack inequality to prove H\"older regularity and existence of solutions of the integro-PDEs.
arxiv
Solutions of nonlinear Sobolev-Burgers PDEs [PDF]
arXiv, 2018 Nonlinear Sobolev-Burgers PDEs are considered. Their solutions are investigated. A technique of noncommutative line integration is utilized for their description. A new method of PDEs solution with the help of Cayley-Dickson algebras is developed in the article. Moreover, random operator valued measures are studied and applied to solutions of PDEs.
arxiv
Hypergeometric functions and the Tricomi operator [PDF]
arXiv, 2003 We show how certain hypergeometric functions play an important role in finding fundamental solutions for a generalized Tricomi operator.
arxiv
Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations
, 2017 We consider the defocusing nonlinear wave equations (NLW) on the two-dimensional torus. In particular, we construct invariant Gibbs measures for the renormalized so-called Wick ordered NLW. We then prove weak universality of the Wick ordered NLW, showing
Oh, Tadahiro, Thomann, Laurent
core
Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms [PDF]
arXiv, 2010 We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions.
arxiv
One-dimensional versions of three-dimensional system: Ground states for the NLS on the spatial grid
, 2018 We investigate the existence of ground states for the focusing Nonlinear Schr\"odinger Equation on the infinite three-dimensional cubic grid. We extend the result found for the analogous two-dimensional grid by proving an appropriate Sobolev inequality ...
Adami, Riccardo, Dovetta, Simone
core
A remark on the history of Hardy inequalities in domains [PDF]
arXiv, 2004 This a very brief account of the main line of development of Hardy inequalities.
arxiv
Well-Posedness of Second-Order Uniformly Elliptic PDEs with Neumann Conditions [PDF]
arXivExtending the results of Nardi (2015), this note establishes an existence and uniqueness result for second-order uniformly elliptic PDEs in divergence form with Neumann boundary conditions. A Schauder estimate is also derived.
arxiv
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