Abstract
In this paper, we prove existence and uniqueness of viscosity solutions to the following system: For \( i\in \left\{ 1,2,\dots ,m\right\} \)
where \( \Omega \subset \mathbb {R}^n \) is a bounded domain, \( \Omega _{L}:=(0,L)\times \Omega \) and \( F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}\) is a general second-order partial differential operator which covers even the fully nonlinear case. (We will call a second-order partial differential operator \(F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}\) fully nonlinear if and only if, it has the following form
with the restriction that at least one of the functional coefficients \( \alpha _{\alpha },\ |\alpha |=2, \) contains a partial derivative term of second order.) Moreover, F belongs to an appropriate subclass of degenerate elliptic operators. Regarding uniqueness, we establish a comparison principle for viscosity sub and supersolutions of the Dirichlet problem. This system appears among others in the theory of the so-called optimal switching problems on bounded domains.
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Notes
Let \( V:[0,L]\times \bar{\Omega }\rightarrow \mathbb {R}^m \) with \( V=( V_1,V_2,\dots , V_m) \) be a bounded function (i.e., \(\text {for all } i\in \{1,2,\dots ,m\},\ \sup \{V_i(y,x)\,\ (y,x)\in [0,L]\times \bar{\Omega }\}<+\infty \) and \( \inf \{V_i(y,x)\,\ (y,x)\in [0,L]\times \bar{\Omega }\}>-\infty \)) and continuous on \( [0,L)\times \bar{\Omega }. \) Then, for each \( i\in \{1,2,\dots ,m\}, \) the following functions
$$\begin{aligned} \hat{V}^1_i(y,x):={\left\{ \begin{array}{ll} V_i(y,x) &{} \ \ (y,x)\in [0,L)\times \bar{\Omega }\\ M &{} y=L,\ x\in \bar{\Omega } \end{array}\right. } \ \text {and} \ \hat{V}^2_i(y,x):={\left\{ \begin{array}{ll} V_i(y,x) &{}\quad (y,x)\in [0,L)\times \bar{\Omega }\\ m &{} y=L,\ x\in \bar{\Omega } \\ , \end{array}\right. } \end{aligned}$$where \( M:=\max _{i\in \{1,2,\ldots ,m\}}\sup \{V_i(y,x)\,\ (y,x)\in [0,L]\times \bar{\Omega }\}\) and \( m:=\min _{i\in \{1,2,\ldots ,m\}}\inf \{V_i(y,x)\,\ (y,x)\in [0,L]\times \bar{\Omega }\} \), are upper semicontinuous and lower semicontinuous on \([0,L]\times \bar{\Omega }\) respectively.
Let \( (X,\rho ) \) be a metric space, \( A\subset X \) and \( f:A\rightarrow [-\infty ,\infty ]. \) We define as lower semicontinuous envelope of the function f the function
$$\begin{aligned} f_{*}:A\rightarrow [-\infty ,\infty ],\ f_{*}(x):=\lim _{\delta \rightarrow 0}\inf \{f(y)\,\ y\in A\cap B_{\rho }(x,\delta )\},\ \text {for all } x\in A. \end{aligned}$$Analogously, we define as upper semicontinuous envelope of the function f, the function
$$\begin{aligned} f^{*}:A\rightarrow [-\infty ,\infty ],\ f^{*}(x):=\lim _{\delta \rightarrow 0}\sup \{f(y)\,\ y\in A\cap B_{\rho }(x,\delta ) \},\ \text {for all } x\in A. \end{aligned}$$We set as a modification of function \( V_i^{i,\hat{x}\epsilon ^*} \) the function \( \tilde{V}_i^{i,\hat{x},\epsilon ^*}:[0,L]\times \bar{\Omega }\rightarrow \mathbb {R}\) defined as \( \tilde{V}_i^{i,\hat{x},\epsilon ^*}(y,x) = {\left\{ \begin{array}{ll} V_i^{i,\hat{x},\epsilon ^*}(t,x) &{}\quad (y,x)\in [0,T)\times \bar{\Omega }\\ M_{\epsilon _o} &{} y=L,\ x\in \bar{\Omega } \\ \end{array}\right. } \), where \( M_{\epsilon _o}=\max _{(y,x)\in [0,L]\times \bar{\Omega }} V_i^{i,\hat{x},\epsilon _o}(y,x), \) which also satisfies \( M_{\epsilon _o}>=\max _{(y,x)\in [0,L]\times \bar{\Omega }} V_i^{i,\hat{x},\epsilon }(y,x),\ \text {for all }\epsilon \in (0,\epsilon _o]. \) Then, the function \( \tilde{V}_i^{i,\hat{x},\epsilon ^*} \) remains upper semicontinuous on its domain and satisfies at the same time \( \tilde{V}_i^{i,\hat{x},\epsilon ^*}\ge V_i^{i,\hat{x},\epsilon *} \) on \( [0,L]\times \bar{\Omega },\ \text {for all }\epsilon >0 \).
Let \( (X,\rho ) \) be a metric space, \( A\subset X,\ x_o\in A \) and \( f,g:A\rightarrow \mathbb {R}\) mappings. If there exists \( \delta _0>0, \) such that \( \text {for all } x\in B_{\rho }\left( x_o,\delta _0 \right) \cap A,\ f(x)=g(x) \) then, \( f^*(x_o)=g^*(x_o) \) and \( f_*(x_o)=g_*(x_o). \)
Let \( (X,\rho ) \) be a metric space, \( A\subset B\subset X,\ x_o\in A \) and \( f:B\rightarrow \mathbb {R}. \) If the following holds
$$\begin{aligned} \exists \delta >0,\ A\cap B_{\rho }(x_o,\delta )=B\cap B_{\rho }(x_o,\delta ) \end{aligned}$$and the restriction \( f|_A \) is continuous on \( x_o \), then the extension f also is continuous on \( x_o. \)
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Acknowledgements
The authors would like to thank the anonymous referee for carefully reading the manuscript and for his/her valuable comments. This work was supported by the University of Cyprus research funds.
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Andronicou, S., Milakis, E. Systems of fully nonlinear degenerate elliptic obstacle problems with Dirichlet boundary conditions. Annali di Matematica 202, 2861–2901 (2023). https://doi.org/10.1007/s10231-023-01343-w
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DOI: https://doi.org/10.1007/s10231-023-01343-w