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Register a new userOn the $sigma_2$-Nirenberg problem on $mathbb{S}^2$
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We establish theorems on the existence and compactness of solutions to the $sigma_2$-Nirenberg problem on the standard sphere $mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Mobius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Mobius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the $sigma_2$-Nirenberg problem, and a B^ocher type theorem for the associated fully nonlinear Mobius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the $sigma_2$-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove $L^infty$ a priori estimates for solutions of the $sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the $sigma_2$-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.
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In this note, we study symmetry of solutions of the elliptic equation begin{equation*} -Delta _{mathbb{S}^{2}}u+3=e^{2u} hbox{on} mathbb{S}^{2}, end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.
We study the problem of prescribing $sigma_k$-curvature for a conformal metric on the standard sphere $mathbb{S}^n$ with $2 leq k < n/2$ and $n geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $beta in [2,n)$. We prove among other things the following statements. (1) When $beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ beta = n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $frac{2}{n-2k}le beta < n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $beta < frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $beta$, there exists also some $K$ for which the solution set is empty.
We consider the energy-critical half-wave maps equation $$partial_t mathbf{u} + mathbf{u} wedge | abla| mathbf{u} = 0$$ for $mathbf{u} : [0,T) times mathbb{R} to mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target $mathbb{S}^2$ applied to half-harmonic maps from $mathbb{R}$ to $mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $mathbf{Q}$ with arbitrary degree $m geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $mathbb{R}$ to the unit circle $mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.
We consider half-harmonic maps from $mathbb{R}$ (or $mathbb{S}$) to $mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is $pm 1$, we prove that the deviation of any map $boldsymbol{u}:mathbb{R}to mathbb{S}$ from Mobius transformations can be controlled uniformly by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. This result resembles the quantitative rigidity estimate of degree $pm 1$ harmonic maps $mathbb{R}^2to mathbb{S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $boldsymbol{u}$ is already near to a Blaschke product, then the deviation of $boldsymbol{u}$ to Blaschke products can be controlled by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $boldsymbol{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${mathbb R}^2to {mathbb S}^2$.
The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.
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