arXiv · math/0501461
We classify homogeneous degree $d\neq2$ solutions to fully nonlinear elliptic equations.
Explore connections, maps & timelines
Nikolai Nadirashvili, Yu Yuan. 2005-01-25. Homogeneous Solutions to Fully Nonlinear Elliptic Equations. https://arxiv.org/abs/math/0501461
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
In this paper, we study the following nonlinear Schrödinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu+V(x)u=\partial_v H(x,u,v)+ωv,\ x \in \mathbb{R}^N, \\ -Δv+V(x)v=\partial_u H(x,u,v)+ωu,\ x \in \mathbb{R}^N, \\ \displaystyle\int_{\R^N}|z|^2dx=a^2, \end{array}\right. \end{equation*} where the potential function $V(x)$ is periodic, $z:=(u,v):\R^N\rightarrow \R\times\R$, $ω\in \R$ appears as a Lagrange multiplier, $a>0$ is a prescribed constant. The existence and multiplicity of $L^2$-normalized solutions for the above Schrödinger system are obtained. The proof combines a constrained variational reduction, compactness estimates, and a Ljusternik--Schnirelmann type multiplicity theorem. In addition, a bifurcation result is also given.
We develop a fully constructive, conservative, and collision-level realization of Kac's program for the spatially homogeneous Landau equation across the full interaction range, including the Coulomb case. Our model is the microcanonical binary-collision (MBC) process: a reversible pure-jump $N$-particle Markov process, realizing the grazing-collision mechanism via small conservative rotations of relative velocities. Under suitable assumptions on the initial data, we prove propagation of chaos in the joint mean-field and grazing-collision limit ($N\to\infty, h\to0$), identifying any limit point with the unique solution in the relevant weak-solution class. The empirical measures converge in quadratic Wasserstein distance, while fixed-order marginals converge strongly in $L^1$ at every fixed time. Furthermore, we rigorously derive the Landau master equation as the fixed-$N$ grazing-collision limit of the MBC process. The proof combines Fisher-information dissipation, compactness, and uniqueness; for $γ\in[-3,-2)$, second-order information along collision directions yields the time-integrable $L^\infty$ bound needed for uniqueness.
In this paper, we investigate the regularization effect of fractional stochastic forcing on Burgers-type equations with fractional dissipation, with an application to the Degasperis--Procesi (DP) equation. In particular, we consider the perturbation induced by the singular noise $|D|^{1/2}ξ$ and establish local well-posedness in the negative Sobolev space $H^{-1/4+δ}$ for some small $δ>0$. Due to the singular nature of the nonlinear interactions, classical solution theories cannot be directly applied. Inspired by the framework developed in \cite{hairer2013solving,gubinelli2017kpz}, we introduce a generalized solution theory based on an enhanced structure and derive the effective equations satisfied by these generalized solutions. Our main contribution is the establishment of a general framework for describing singular PDEs driven by rough data. Moreover, we prove the convergence of the associated non-Gaussian rough structures in the fractional dissipation setting. As an application, we apply this framework to the stochastic Degasperis--Procesi equation and obtain its local well-posedness in the low-regularity regime.