Dirichlet problems involving the 1-Laplacian
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Dirichlet problems involving the 1-Laplacian Yawei Wei1
· Huiying Zhao2
Received: 2 March 2020 / Revised: 30 April 2020 / Accepted: 5 May 2020 © Springer Nature Switzerland AG 2020
Abstract The present paper concerns Dirichlet problem for a class of non-linear partial differential equations involving 1-Laplacian. The non-linear term holds the sub-critical exponential growth and guarantees the solution is non-trivial. The difficulty provided by the degeneration of 1-Laplacian has been overcame by a replacement with a suitable vector field. The underlying minimization problem has been investigated to verify the existence result to the concern problem. Keywords 1-Laplacian · Dirichlet problems · Non-linear equations · Non-smooth analysis Mathematics Subject Classification 35A01 · 35J60 · 35J70
1 Introduction In this paper, we are interested in the existence of a non-trivial solution to the following Dirichlet problem
−1 u(x) = f (x, u, g), u(x) = 0,
x ∈ , x ∈ ∂.
(P)
Here is an open bounded domain with Lipschitz boundary in R N and N ≥ 2 with its closure and boundary are separately denoted as and ∂. The involving operator 1-Laplacian is defined as
B
Yawei Wei [email protected] Huiying Zhao [email protected]
1
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
2
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
Y. Wei, H. Zhao
1 =: Div
Du . |Du|
(1)
The nonlinear function f (x) includes a given function g(x). The 1-Laplacian defined in (1) is not well defined wherever |Du| is zero, which can be seen as the formal limit of p-Laplacian, as p → 1+ , see [12]. Therefore, problems involving 1-Laplacian in bounded domain can be studied via p-Laplacian problems, and then taking the limit p → 1+ , as in [10,12]. Beside this approximation approach, Du by a well defined vector field Andreu et al. [2,4] replaced the degeneration part |Du| which extends the former one wherever Du vanishes. The Dirichlet problem has also been investigated in [2], here since the Dirichlet boundary condition does not hold in the usual trace form, they also introduce a weak formulation to deal this problem. Here we also recall that early papers devoted to the Dirichlet problem of equations involving the 1-Laplacian, including [2,3,5,11,14,17,18]. The interest in this setting comes from an optimal design problem in the theory of torsion and from the level set formulation of the inverse mean curvature flow. K.C. Chang investigates the spectrum of 1-Laplacian in [6]. On the other hand, 1-Laplacian problems also appear in game theory, see [16] and the variational approach to image restoration, see [7]. Indeed, the total variation minimizing models have become one of the most popular and successful methodology for image restoration since the introduction of the ROF model by Rudin et al. [20]. The present paper holds the following results. Theorem 1 Assume that f (x, u, g) satisfies (f1) f (x, u, g) ∈ C( × R × R) and for some 1 < r < 1∗ , c > 0 | f (x
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