arXiv research

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

Smooth solutions found for a specific type of Yamabe problem.

Maps in Carnot groups are equivalent to solutions of a PDE system.

Study properties of solutions with singularities in the negative cone.

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

New MIP formulations for neural network Lipschitz constant estimation.

We prove that Lipschitz intrinsic graphs in the Heisenberg groups $H^n$, with $n>1$, which are vanishing viscosity solutions of the minimal surface equation are smooth.

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…

New methods solve MI problems with locally Lipschitz operators, improving solution efficiency.

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Further…

In this paper, we provide conditions which ensure that stochastic Lipschitz BSDEs admit Malliavin differentiable solutions. We investigate the problem of existence of densities for the first components of solutions to general path-dependent stochastic Lipschitz BSDEs and obtain results for the second components in part…

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean s…

New parameterization of neural networks with Lipschitz bounds for robustness.

New method proves heat flow of harmonic maps into CAT(0) spaces.

Maximal causal curves for Lipschitz metrics are either lightlike or timelike.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

Proves continuum limits of Lipschitz learning using Γ-convergence.

By using Bäcklund transformation for the sine-Gordon equation, new periodic exact solutions of the constant astigmatism equation $z_{yy} + ({1}/{z})_{xx} + 2 = 0$ are generated from a seed which corresponds to Lipschitz surfaces of constant astigmatism.

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spac…

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for o…

Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.

Study on curvature equation in Heisenberg group with convex boundary.

A new algorithm solves minimax problems without needing parameters.

We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingale…

New framework enhances neural network robustness against adversarial attacks.

We prove that the geodesic equation for any semi-Riemannian metric of regularity $C^{0,1}$ possesses $C^1$-solutions in the sense of Filippov.

Study on the nodal set of Dirac equation solutions on manifolds.

New estimates for nodal and singular sets of parabolic inequalities.

Efficient binary sampling method for global optimization of univariate functions with low regret.

Study invariant Lipschitz bandits, improving regret bounds.

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

Probabilistic learning is increasingly being tackled as an optimization problem, with gradient-based approaches as predominant methods. When modelling multivariate likelihoods, a usual but undesirable outcome is that the learned model fits only a subset of the observed variables, overlooking the rest. In this work, we …

Study on free boundary problems in RCD spaces, proving existence and regularity.

MLDL preserves manifold geometry in vector transformations.

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

The paper proves Lipschitz regularity of graph Laplacian eigenvectors on random data clouds.

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing spars…

In this paper, we study the valuation of American type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condi…

Neural operators learn to solve LQ MFGs efficiently in infinite dimensions.

We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipsc…

New methods for convex optimization with locally Lipschitz gradient, achieving faster convergence.

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded as a hypersurface of some Euclidean space and the initial condition is a Lipschi…

We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela …

We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological …

In this paper, we investigate the underlying factor that leads to failure and success in the training of GANs. We study the property of the optimal discriminative function and show that in many GANs, the gradient from the optimal discriminative function is not reliable, which turns out to be the fundamental cause of fa…

Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.

We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output relation. To that end, we first establish a global bound for the Lipschitz constant of …