Quasilinear second order elliptic equations with Venttsel boundary conditions

A survey of results on nonlinear Venttsel problems

“…The elliptic analogue of Theorem 1.2 can also be found in [6] (see Theorem 1 ). It generalizes the results of [20][21] to the case of unbounded b, β, Φ 1 and Θ 1 . Then…”

“…The corresponding results for the stationary problem were obtained in [7]. Note that the gradient estimates for solutions of stationary problems were established in [21] under the hypotheses more limiting than ours: the right-hand sides of the equation and the boundary condition were assumed to be differentiable with respect to all variables, the function α(x, z, p) had to be nondegenerate with respect to the normal component of the gradient and could have at most the linear growth according to p. Note also that the arguments from [7,9] can be extended to the case of fully nonlinear Venttsel boundary value problems.…”

“…The classical solvability of the problem for a quasilinear uniformly elliptic equation with a quasilinear uniformly elliptic Venttsel condition was established by Y. Luo and N. S. Trudinger in the papers [20][21][22]. This result was generalized in [7].…”

Regularity for steady periodic capillary water waves with vorticity

“…then from (2.19) and (3.6), we have that T + lv p is a non-degenerate, uniformly oblique Venttsel-type boundary operation [12,13]. It follows directly from the results of §5 in the study of Walsh [13] that D 1 F(h 0 , 0) is a Fredholm operator of index zero.…”

“…Here, the presence of surface tension is a significant complicating factor (cf. [11][12][13][14] and below) in the analysis and it is perhaps a miracle of the inherent structure of the water wave equations that such results carry over to this case. The analyticity of the free surface for capillary-gravity waves when the vorticity function is analytic was established in Henry [15].…”