By leveraging an inherently geometric argument, we establish new properties of convex functions Γ defined on a convex domain Ω . Suppose the graph of the convex function Γ contains a line segment [Y^1, Y^2] , where Y^j = (y^j, Γ (y^j)) , with y^j ∈Ω , and that Y^* = (y^*, Γ (y^*)) lies on this segment. Given a second-order polynomial P, whose graph touches (locally) the graph of Γ from below at Y^* , then a horizontal/vertical translation of P/3 touches the convex graph of Γ at least at one of the points Y^j . This, in light of the viscosity approach in PDEs, has interesting consequences for the regularity of convex envelopes of supersolutions to a large class of partial differential equations. These include degenerate fully nonlinear models and quasi-linear problems that have not been treated in the literature. Our methods are versatile, and we expect them to find applications in a broader class of models.
In this work, we investigate a two-species chemotaxis system with singular sensitivity and indirect signal production in an n-dimensional domain (n≥ 1) . The movement of each species is governed by a nonlinear sensitivity function that responds to chemical gradients. The chemical substances interact dynamically: one chemical is secreted by the species, while the other degrades over time and influences the movement. The model also incorporates logistic growth and interspecific competition between the species. Using semigroup theory and a Moser-type iteration framework, we establish the global boundedness of classical solutions.
In this paper, we first derive area estimates for weighted stable minimal capillary surfaces in a 3-dimensional weighted Riemannian manifold M with boundary ∂M , in terms of the weighted scalar curvature of M and the weighted mean curvature of ∂M . As an application of these estimates, we establish a rigidity theorem for 3-dimensional weighted Riemannian manifolds in the equality case. Finally, we investigate the relationship between the weighted scalar curvature of M (or the weighted mean curvature of ∂M ) and the topological properties of weighted stable minimal capillary surfaces.
Abstract In this paper we present a new global $${L^\infty }$$ L ∞ -estimate for solutions $$u\in D^{s,p}({\mathbb R}^N)$$ u ∈ D s , p ( R N ) of the fractional p -Laplacian equation $$ u\in D^{s,p}({\mathbb R}^N): (-\Delta _p)^s u=f(x,u) \quad \text{ in } {\mathbb R}^N, $$ u ∈ D s , p ( R N ) : ( - Δ p ) s u = f ( x , u ) in R N , of the form $$ \Vert u\Vert _{\infty }\le C \Phi (\Vert u\Vert _{\beta }) $$ ‖ u ‖ ∞ ≤ C Φ ( ‖ u ‖ β ) for some $$\beta > p$$ β > p , where $$\Phi : {\mathbb R}^+\rightarrow {\mathbb R}^+$$ Φ : R + → R + is a data independent function with $$\lim _{s\rightarrow 0^+}\Phi (s)=0$$ lim s → 0 + Φ ( s ) = 0 . The obtained $$L^\infty $$ L ∞ -estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the $$L^\infty $$ L ∞ and decay estimate we prove a Brezis-Nirenberg type result regarding $$D^{s,2}({\mathbb R}^N)$$ D s , 2 ( R N ) versus $$C_b\left( {\mathbb R}^N, 1+|x|^{N-2s}\right) $$ C b R N , 1 + | x | N - 2 s local minimizers.
