arXiv research

Deep neural nets solve complex insurance math equations.

We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.

Study shows solutions to certain equations form smooth manifolds.

The paper develops Morse homology for a class of elliptic partial differential equations.

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…

Paper classifies minimal graph transformations into new families of surfaces.

We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of C…

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine para…

The paper examines ellipticity of specific equations on vector bundles.

Solves division problem for L. Hörmander's systems.

New theorems prove uniqueness of solutions to geometric PDEs.

Survey of geometry developments, including complex structures on surfaces.

New method for analyzing elliptic and parabolic equations.

Uniform estimates for complex equations on compact manifolds found.

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $\frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of M…

Perimeter on manifolds leads to new symmetrization methods.

Rust library solves complex equations on abstract simplicial complexes.

The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.

Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half-space. We establish e…

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_Ωφ\left( \left\vert \nabla u\right\vert \right) , \] where $Ω$ is a bounded domain of a complete Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Ω=\mathcal{M}$ is a C…

These are lecture notes for the mini-course \textit{PDE and hypersurfaces with prescribed mean curvature} held in Federal University of São Carlos at the Workshop on Submanifold Theory and Geometric Analysis, August 05 -- 09, 2019. The aim of these notes is to introduce to the geometers useful tools from the \textit{Th…

In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conforma…

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a min…

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nab…

We show that a subspace $S$ of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are that $S$ is closed in $L^2(M)$ and that if a sequence of functions $f_n$ in $S$ …

Unified approach combining BSDEs and PINNs for solving PDEs.

This paper bounds the volume of singular and critical sets for elliptic equations with Hölder coefficients.

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is giv…

`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing tec…

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate ell…

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm of the second fundamental form and satisfying so-called `flat boundary conditio…

Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.

Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.

Rigidity for 4D Willmore submanifolds with boundary.

We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate to Lie groups. We then consider the static case, which describes crystalline bodies in equilibriu…

Distance between evolving hypersurfaces is a PDE solution.

The paper provides estimates for eigenvalues of elliptic differential problems.

Study proves Maximum Principles for unbounded Riemannian domains.

We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over th…

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…

Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouvil…

PILNO uses neural operators to solve PDEs efficiently on point clouds.