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Register a new userOn the Liouville Theorems for infinity Laplacian with gradient and KPP type equation
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In this paper, we prove several Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + f(x, u),=,0quad text{in}; mathbb{R}^d,$$ where $gammain [0, 2]$ and $Delta^gamma_infty$ is a $(3-gamma)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $liminf_{|x|toinfty}lim_{sto0}f(x,s)/s^{3-gamma}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-gamma}$ decreasing in $s$, we also obtain the uniqueness. While, if $limsup_{|x|toinfty}sup_{[delta_1,delta_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop new technique to overcome the degeneracy of infinity Laplacian and nonlinearity of gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopfs lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further study the related Dirichlet principal eigenvalue of the corresponding nonlinear operator $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + c(x)u^{3-gamma},$$ in smooth bounded domains, which may be considered as of independent interest. Our results could be seen as the extension of Liouville type results obtained by Savin [48] and Ara{u}jo et. al. [1] and a counterpart of the uniqueness obtained by Lu and Wang [39,40] for sign-changing $f$.
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In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.
We propose two asymptotic expansions of the two interrelated integral-type averages, in the context of the fractional $infty$-Laplacian $Delta_infty^s$ for $sin (frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland-Caffarelli-Figalli, 2012]. Our expansions are parametrised by the radius of the removed singularity $epsilon$, and allow for the identification of $Delta_infty^sphi(x)$ as the $epsilon^{2s}$-order coefficient of the deviation of the $epsilon$-average from the value $phi(x)$, in the limit $epsilonto 0+$. The averages are well posed for functions $phi$ that are only Borel regular and bounded.
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak cite{KNSS}, to the MHD setting.
We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${mathcal L_0}$ in $mathbb{R}^N$, as a consequence of a Liouville theorem at $t=- infty$ for the corresponding Kolmogorov operators ${mathcal L_0} - partial_t$ in $mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({mathcal L_0} - partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.
In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $|f{u_r}{r}{bf 1}_{{u_r< -f 1r}}|_{L^{3/2}(mbR^3)}< C_{sharp}$ where $C_{sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)geq -f1r$ for $forall (r,z)in[0,oo)timesmbR$, then ${bf u}equiv 0$. Liouville theorems also hold if $displaystylelim_{|x|to oo}Ga =0$ or $Gain L^q(mbR^3)$ for some $qin [2,oo)$ where $Ga= r u_{th}$. We also established some interesting inequalities for $Omco f{p_z u_r-p_r u_z}{r}$, showing that $ aOm$ can be bounded by $Om$ itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with ${bf u}=u_r(r,z){bf e}_r +u_{th}(r,z) {bf e}_{th} + u_z(r,z){bf e}_z, {bf h}=h_{th}(r,z){bf e}_{th}$, indicating that the swirl component of the magnetic field does not affect the triviality. Especially, we establish the maximum principle for the total head pressure $Phi=f {1}{2} (|{bf u}|^2+|{bf h}|^2)+p$ for this special solution class.
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