Abstract The Chaplygin gas model is both interesting and important in the theory of gas dynamics and conservation laws, all the characteristic families of which are linearly degenerate. Majda conjectured that the shock formation never happens for smooth data. In this article, we prove the conjecture for the two space dimensional axisymmetric case. Different from previous approaches to study wave equations with different speeds, we reformulate the problem in the Lagrangian coordinates and consider a single wave equation with variable coefficients. This not only gives a simpler proof but also enables us to treat the case with large angular velocities.

Abstract In this article, we establish a Gaffney type inequality, in W ℓ , p {W}^{\ell ,p} -Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p ∈ ] 1 , ∞ [ p\in ]1,\infty {[} and ℓ = 1 , 2 \ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

Abstract In this article, we investigate the boundary blow-up problem Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{\infty }^{h}u=f\left(x,u),& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=\infty ,& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ {\Delta }_{\infty }^{h}u=| Du\hspace{-0.25em}{| }^{h-3}\langle {D}^{2}uDu,Du\rangle is the highly degenerate operator related to infinity Laplacian which comes from the absolutely minimizing Lipschitz extension and has a close relationship with the random game named tug-of-war. When the function f f satisfies the Keller-Osserman-type condition, we establish the existence of the boundary blow-up viscosity solution. Moreover, for the separable case f ( x , u ) = b ( x ) g ( u ) f\left(x,u)=b\left(x)g\left(u) , we establish the asymptotic estimate of the blow-up solution near the boundary under some regular conditions of the domain. Based on the asymptotic estimate and comparison principle, we obtain the uniqueness of the large viscosity solution. During this procedure, we also study the non-existence of the large solution. For the separable case, we show that the Keller-Osserman-type condition is sufficient and necessary for the existence of the boundary blow-up viscosity solution.

Abstract We consider Monge-Ampère type equations involving the gradient that are elliptic in the framework of convex functions. Through suitable symmetrization we find sharp estimates to solutions of such equations. An overdetermined problem related to our Monge-Ampère type operators is also considered and we show that such a problem may admit a solution only when the underlying domain is a ball.

Abstract The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

Abstract In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle% \phantom{}\text{in }\mathbb{R}^{3},\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right. where ϵ > 0 {\epsilon>0} , a , b > 0 {a,b>0} are constants, V : ℝ 3 → ℝ {V:\mathbb{R}^{3}\rightarrow\mathbb{R}} and A : ℝ 3 → ℝ 3 {A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}} are continuous potentials, and Δ A ⁢ u {\Delta_{A}u} is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Abstract In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem.

Abstract The aim of this paper is analyzing the positive solutions of the quasilinear problem-(u′/1+(u′)2)′=λ⁢a⁢(x)⁢f⁢(u) in ⁢(0,1),u′⁢(0)=0,u′⁢(1)=0,-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,whereλ∈ℝ{\lambda\in\mathbb{R}}is a parameter,a∈L∞⁢(0,1){a\in L^{\infty}(0,1)}changes sign once in(0,1){(0,1)}and satisfies∫01a⁢(x)⁢𝑑x<0{\int_{0}^{1}a(x)\,dx<0}, andf∈𝒞1⁢(ℝ){f\in\mathcal{C}^{1}(\mathbb{R})}is positive and increasing in(0,+∞){(0,+\infty)}with a potential,F⁢(s)=∫0sf⁢(t)⁢𝑑t{F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at+∞{+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions,𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from(λ,0){(\lambda,0)}at someλ0>0{\lambda_{0}>0}and from(λ,∞){(\lambda,\infty)}at someλ∞>0{\lambda_{\infty}>0}. It also establishes that𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}consists of regular solutions if and only if∫0z(∫xza(t)dt)-1/2dx=+∞ or ∫z1(∫xza(t)dt)-1/2dx=+∞.\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty.Equivalently, the small positive regular solutions of𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}become singular as they are sufficiently large if and only if(∫xza(t)dt)-1/2∈L1(0,z) and (∫xza(t)dt)-1/2∈L1(z,1).\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1).This is achieved by providing a very sharp description of the asymptotic profile, asλ→λ∞{\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions ofλ0{\lambda_{0}}andλ∞{\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.

Abstract We consider the high-dimensional equation ∂ t ⁡ u - Δ ⁢ u m + u - β ⁢ χ { u > 0 } = 0 {\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0} , extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution u ∈ 𝒞 ⁢ ( [ 0 , T ] ; L δ 1 ⁢ ( Ω ) ) {u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))} , with u - β ⁢ χ { u > 0 } ∈ L 1 ⁢ ( ( 0 , T ) × Ω ) {u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)} , δ ⁢ ( x ) = d ⁢ ( x , ∂ ⁡ Ω ) {\delta(x)=d(x,\partial\Omega)} , we prove some pointwise gradient estimates for a certain range of the dimension N, m ≥ 1 {m\geq 1} and β ∈ ( 0 , m ) {\beta\in(0,m)} , mainly when the absorption dominates over the diffusion ( 1 ≤ m < 2 + β {1\leq m<2+\beta} ). In particular, a new kind of universal gradient estimate is proved when m + β ≤ 2 {m+\beta\leq 2} . Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.