Nonlinear boundary conditions for nonlinear second order differential operators on c[0,1]


Of concern is the operator script A sign defined formally by (script A sign u)(x) = φ((x,u′(x))u″(x) + ψ(x, u(x), u′(x)), with certain assumptions on φ, ψ. Realizations A of script A sign will be definedon C[0,1] and determined by boundary conditions. In earlier work we look ψ ≡ 0 and used mixed Wentzell-Robin boundary conditions of the form aj(Au)(j) + bju′(j) + cju(j) = 0, j = 0,1 where (aj,bj,cj) ≠ (0,0,0). In this paper we consider more general boundary conditions of the form αj(Au)(j) + βju′(j) ∈ γj(u(j)) where γj is a maximal monotone graph in ℝ2 and (αj,βj) ≠ (0,0). The conclusion (under suitable hypotheses) is that A is m-dissipative and Cauchy problem du/du = script A sign u, u(0) + f is well-posed.

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Archiv der Mathematik