arXiv subjects

Ram Baran Verma

Publications and source records attributed to Ram Baran Verma.

11 recordsLinked to original sources

arXiv2026

Estimates on elliptic equations that hold only where the Hessian is large

In this article, we establish Hölder regularity for viscosity solutions to a class of degenerate fully nonlinear elliptic equations of the form \[ F(D^2u,Du)=f(x)~~\text{in}~~B_1, \] where the operator is elliptic only in regions where the Hessian is sufficiently large. Such equations arise naturally in free boundary problems and models with partial ellipticity. The proof combines a modified cusp function with a decomposition of the contact set in a point-to-measure argument. As a consequence, interior Hölder continuity follows under natural structural assumptions.

math.AP↗

arXiv2026

Shape Optimization for the Principal Eigenvalue of the Pucci Operator in Three Dimensions

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{λ,Λ}(D^{2}u)&=μ^{+}_{1}(Ω)u &&\text{in }Ω,\\ u &=0 &&\text{on }\partialΩ, \end{aligned} \right. \] in dimension three. Since $\mathcal{M}^+_{λ,Λ}$ is fully nonlinear, in non-divergence form, and non-variational, classical symmetrization and rearrangement methods are not available. We introduce a three-dimensional family of double--pyramidal domains $\{Ω^ω_{γ,a}\}$ parametrized by an anisotropy factor $γ\in \left[\frac{1}{\sqrtω},\sqrtω\right]$ and an affine shear parameter $a\in(-π,π)$, under fixed ellipticity ratio $ω=Λ/λ\ge 1$. Within this family and under a fixed-volume constraint, we prove that the volume-normalized principal eigenvalue is uniquely minimized at the symmetric unsheared configuration $(γ,a)=(1,0)$ among domains in the family $\{Ω^ω_{γ,a}\}$. The proof combines an explicit construction of positive eigenfunctions on seven patches with a lower bound under affine shear deformations. Using the homogeneity and orthogonal invariance of the Pucci operator, we identify an involutive symmetry $γ\mapsto γ^{-1}$ in the associated volume functional and establish strict monotonicity away from the self-dual point $γ=1$. In particular, for $ω>1$, any nontrivial anisotropy or shear strictly increases the normalized principal eigenvalue. This reveals a genuinely three-dimensional rigidity mechanism for a fully nonlinear spectral problem and extends to dimension three the symmetry-minimization phenomenon previously known in the planar case.

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arXiv2026

Liouville type theorems for fully nonlinear elliptic equations with superlinear growth in gradient

This article investigates positive supersolutions of the fully nonlinear elliptic system \[ \begin{cases} -\mathcal{M}_{λ,Λ}^{+}(D^{2}u)+|\nabla u|^{q} \geqλ_{1}f_{1}(v)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0},\\ -\mathcal{M}_{λ,Λ}^{+}(D^{2}v)+|\nabla v|^{q} \geqλ_{2}f_{2}(u)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0}, \end{cases} \] where $q>1,λ_{1},λ_{2}>0,$ and the nonlinearities exhibit power-type behaviour either near the origin or at infinity. Introducing the effective dimension $\widetilde n_{+}=\fracλΛ(n-1)+1$ associated to extremal Pucci operator, we identify the critical exponent $q_{c}=\frac{\widetilde n_{+}}{\widetilde n_{+}-1},$ which governs the qualitative behaviour of positive supersolutions. Using this framework, we establish sharp Liouville-type nonexistence theorems in exterior domains and determine optimal nonexistence regions through the interaction between the gradient exponent $q,$ the effective dimension $\widetilde n_{+}$ and the nonlinear couplings. In the prototype case $f_{1}(t)=t^{p_{1}},$ $f_{2}(t)=t^{p_{2}}$ the obtained conditions are shown to be optimal. The analysis is carried out under the natural regularity assumption $u,v\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus B_{R_0}),$ for $p>n,$ which is the regularity available for fully nonlinear uniformly elliptic equations, rather than within a classical $C^{2}$ framework. Our results provide the fully nonlinear Pucci analogue of the Liouville theory for semilinear elliptic systems involving nonlinear gradient terms and reveal the fundamental role of the effective dimension in determining the critical nonexistence thresholds.

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arXiv2026

Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator

Consider \[ \begin{cases} F(D^2 u,Du,u,x) = u^{-p}v^{-q},~\text{in}~Ω\\ F(D^2 v,Dv,v,x)=u^{-r}v^{-s},~~\text{in}~~Ω\\ u,v>0~~\text{in}~~Ω\\ u=v=0~\quad~\text{on}~~\partialΩ, \end{cases} \] where $Ω$ is an open connected subset of $\mathbb{R}^{N}$ and $p,s$ are two non-negative and $q,r$ are positive real numbers. This article discuses the conditions in terms of the relations among $p,q,r$ and $s$ which lead to existence, uniqueness and non-existence of positive solutions to the system. Furthermore, we also have studied some regularity properties of solution of the system. These results are inspired by the study of Lane-Emden system of equations as in \cite{busca2002liouville,ghergu2010lane}.

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arXiv2024

Exceptional Boundary Sets for Solutions of Fully Nonlinear Parabolic PDEs

This article investigates the exceptional set of the boundary for the following problem: \begin{equation*} \begin{aligned} -\frac{\partial u}{\partial t} + \mathcal{M}_{λ,Λ}^+(D^2u) + b(x,t)\cdot Du + c(x,t)u =0 \quad \rm{in} ~ Ω_{T}, \end{aligned} \end{equation*} We provide a sufficient condition on the exceptional set in terms of the bound of the Hausdorff measure of this boundary portion. This condition ensures that even if the boundary values are not nonnegative on this portion, the supersolution remains nonnegative.

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arXiv2024

Regularity for Fully Nonlinear Elliptic Equations with Natural Growth in Gradient and Singular Nonlinearity

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-α}&=0~\text{in}~Ω\\ u&=0~~\text{on}~~\partialΩ, \end{aligned} \right. \end{equation*} where $Ω$ is a bounded and $C^{2}$ smooth domain in $\mathbb{R}^N$ and $F$ has superlinear growth in gradient and $c(c)<-c_{0}$ for some positive constant $c_{0}.$ Here, we studies the boundary behaviour of the solutions to above equation and establishes the global regularity result similar to one established in [12,16] with linear growth in gradient.

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arXiv2024

Multiplicity results for fully nonlinear elliptic equations with natural gradient growth

In this paper, we prove a theorem concerning the existence of three solutions for the following boundary value problem: \begin{equation*} -\mathcal{M}_{λ,Λ}^+(D^2u)-Γ|Du|^2=f(u)~~~\text{in}\ Ω, u=0~~~\text{on}\ \partialΩ, \end{equation*} where $f:[0,\infty]\to[0,\infty]$ is a $C^α$ function and $Ω$ denotes a bounded, smooth domain in $\mathbb{R}^N$. By constructing two ordered pairs of sub and supersolutions for a specific class of $f$ exhibiting sublinear growth, we further establish the existence of three positive solutions to the aforementioned boundary value problem.

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arXiv2019

Borderline gradient estimates at the boundary in Carnot groups

In this article, we prove the continuity of the horizontal gradient near a $C^{1,\text{Dini}}$ non-characteristic portion of the boundary for solutions to $Γ^{0, \text{Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below where the scalar term is in scaling critical Lorentz space $L(Q,1)$ with $Q$ being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the $Γ^{1, α}$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.

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arXiv2019

$C^{1, α}$ Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions

In this paper, we establish $C^{1, α}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, α}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.

math.AP↗

arXiv2019

Twice differentiability of solutions to fully nonlinear parabolic equations near the boundary

In this paper, we prove $\mathcal{H}^{2+α}$ regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar $C^{2, α}$ regularity result due to Silvestre and Sirakov proved in [15] for solutions to non-convex fully nonlinear elliptic equations.

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arXiv2016

On the bifurcation for fractional Laplace equations

In this paper, we consider the bifurcation problem for fractional Laplace equation \begin{eqnarray*} \begin{array}{ll} (-Δ)^{s} u = λu + f(λ,\,x,\,u)& \mbox{in }Ω, u = 0 &\mbox{in }\mathbb{R}^n\backslash Ω, \end{array} \end{eqnarray*} where $Ω\subset \mathbb{R}^n,\,n> 2s (0<s<1)$ is an open bounded subset with smooth boundary, $(-Δ)^{s}$ stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue $λ_1$ of the eigenvalue problem \begin{eqnarray*} \begin{gathered} (-Δ)^{s} v = λv\,\,\,\mbox{in}\,\,Ω, v = 0 \,\,\,\,\mbox{in}\,\,\,\,\mathbb{R}^n \backslashΩ, \end{gathered} \end{eqnarray*} and, conversely.

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