Existence and Uniqueness of a Weak Solution to a Nonlocal Semilinear Elliptic Problem with Robin Boundary Conditions
Keywords:
Semilinear elliptic equation, nonlocal Robin condition, Lax-Milgram theorem, Banach fixed-point theorem, finite difference methodAbstract
This paper addresses a one-dimensional semilinear elliptic equation with a nonlocal Robin condition on the boundary. The problem is formulated precisely using a suitable Sobolev space, namely a Hilbert space. Due to the presence of the nonlinear boundary, the Lax-Milgram theorem cannot be directly applied to the problem. To overcome this problem, we freeze the problem by fixing the nonlinearity. For the resulting linear problem, we prove the existence of a small weak solution using the Lax-Milgram theorem, which defines the solution operator. Under the small Lipschitz condition on the nonlinearity, we also show that the operator is a contraction. The Banach fixed-point theorem guarantees the existence of a unique fixed point, which is the weak solution to the original problem. Finally, we present a numerical example using the finite difference method to support the theoretical results.
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