Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

Abstract

It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of R n with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product M n × R. Precisely, given a C 2 bounded domain Ω in M and a function H = H(x, z) continuous in Ω × R and nondecreasing in the variable z, we prove that the strong Serrin condition (n − 1)H ∂Ω (y) ≥ n sup z∈R |H(y, z)| ∀ y ∈ ∂Ω, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins-Serrin and Serrin type sharp solvability criteria. * Supported by CAPES and CNPq of Brazil. † Partially supported by CNPq of Brazil. 2000 AMS Subject Classification: 53C42, 49Q05, 35J25, 35J60.

Existence Serrin type results for the Dirichlet problem for the prescribed mean curvature equation in Riemannian manifolds

“…10 p. 6]). We have also proved that (7) is sharp in every compact and simply connected manifold which is strictly 1/4−pinched 2 provided diam(Ω) < π 2 √ K 0 (see [3,Corollary 3 p. 4]). A further Serrin type solvability criteria is directly obtained by combining this non-existence result with Theorem 3.…”

“…On the other hand, in a previous work we have proved that the strong Serrin condition (7) is sharp in every Hadamard manifold (see [3,Corollary 2 p. 3]). The combination of this non-existence result with Theorem 3 fully generalizes Theorem 1 to every Hadamard manifold in the C 2,α class (see also [3,Th. 10 p. 6]).…”

“…We note that our non-existence result for Hadamard manifolds [3, Cor. 1 p. 3] guaranties that Theorem 4 is sharp (see also [3,Th. 7 p. 5]).…”

“…Besides, putting together Spruck's existence theorem [20,Th. 1.4 p. 787] one can deduce that the Serrin sharp solvability criterion for constant H stated in Theorem 2 also holds in the hyperbolic space (see also [3,Th. 8…”

Analyzing the Solution of Chemotaxis Equations with Logistic Source Term

“…In the above equations, Ω ⊂ * 3 is the bounded region with smooth boundary, n is the normal vector outside the unit on zΩ, μ ∈ * , σ > 0. x t represents the trend primary equation under the influence of time series parameters; z t represents the dependent variable affected by the result of the equation; Δx and Δz represent the difference value, respectively; T represents the periodic parameter; u represents the variable of the trend equation. e global existence and boundedness of initial boundary value solutions are discussed [13]. Assume that the initial value meets the conditions shown in the following equation:…”