Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and inequalities of the form:

$$ f_i(z_1,\ldots,z_m) = 0, \text{ for $i = 1,\ldots,k$} $$ and $$ g_j(z_1, \ldots, z_m) \geq 0, \text{ for $j = 1,\ldots,l$}, $$

where the $f_i$ and $g_j$ are real-valued real analytic functions.

Consider the problem of minimizing $u$ restricted to $S$.

Here is what I am hoping for. I am hoping to be able to prove, under some hypotheses, that

$$ \operatorname{min} \{ u(\mathbf{z}) ; \mathbf{z} \in S \}, $$

(where $\mathbf{z} = (z_1,\ldots,z_m)$) is attained at some point in $\partial S$.

Here is an idea, but I am not sure if it will work. Convert the constrained optimization problem to an unconstrained one, for instance using Lagrange multipliers, and then try to apply a weak minimum principle. I will have to think more about it. One issue is that the real analytic subset $S$ is real, and not complex, so I will have to think under what conditions my approach will work. In case such ideas ring a bell, and someone has come across an article along these lines, please inform me.

Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and inequalities of the form:

$$ f_i(z_1,\ldots,z_m) = 0, \text{ for $i = 1,\ldots,k$} $$ and $$ g_j(z_1, \ldots, z_m) > 0, \text{ for $j = 1,\ldots,l$}, $$

where the $f_i$ and $g_j$ are real parts of corresponding holomorphic functions on $U$. Assume that $\bar{S} \subseteq U$ and that $\bar{S}$ is compact.

Consider the problem of minimizing $u$ restricted to $\bar{S}$.

Here is what I am hoping for. I am hoping to be able to prove, under some hypotheses, that

$$ \operatorname{max} \{ u(\mathbf{z}) ; \mathbf{z} \in \bar{S} \}, $$

(where $\mathbf{z} = (z_1,\ldots,z_m)$) cannot be attained at a point in $S$ (the interior of $\bar{S}$) unless $u$ is constant. In other words, I am hoping that the strong maximum principle holds in this setting (perhaps under some hopefully mild hypotheses).

Here is an idea, but I am not sure if it will work. Convert the constrained optimization problem to an unconstrained one, for instance using Lagrange multipliers, and then try to apply the strong maximum principle. I will have to think more about it.

Has the problem of minimizing the real part of a holomorphic function (thus of a pluriharmonic function) on a complex manifold restricted to a real smooth (let us say compact, to simplify) submanifold with boundary been studied before? What kind of inequalities may hold for instance?

Here is what I am hoping for. I am hoping to find a lower bound of the function I am minimizing in terms of the values of the function at the boundary. So I am hoping for some kind of weak minimum principle in this setting, under some hypotheses of course.

Here is an idea, but I am not sure if it will work. Convert the constrained optimization problem to an unconstrained one, for instance using Lagrange multipliers, and then try to apply a weak minimum principle. I will have to think more about it. One issue is that the submanifold is real, and not complex, so I will have to think under what conditions my approach will work. In case such ideas ring a bell, and someone has come across an article along these lines, please inform me.

Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and inequalities of the form:

$$ f_i(z_1,\ldots,z_m) = 0, \text{ for $i = 1,\ldots,k$} $$ and $$ g_j(z_1, \ldots, z_m) \geq 0, \text{ for $j = 1,\ldots,l$}, $$

where the $f_i$ and $g_j$ are real-valued real analytic functions.

Consider the problem of minimizing $u$ restricted to $S$.

Here is what I am hoping for. I am hoping to be able to prove, under some hypotheses, that

$$ \operatorname{min} \{ u(\mathbf{z}) ; \mathbf{z} \in S \}, $$

(where $\mathbf{z} = (z_1,\ldots,z_m)$) is attained at some point in $\partial S$.

Here is an idea, but I am not sure if it will work. Convert the constrained optimization problem to an unconstrained one, for instance using Lagrange multipliers, and then try to apply a weak minimum principle. I will have to think more about it. One issue is that the real analytic subset $S$ is real, and not complex, so I will have to think under what conditions my approach will work. In case such ideas ring a bell, and someone has come across an article along these lines, please inform me.

Has the problem of minimizing the real part of a holomorphic/meromorphic function on a Kähler manifold restricted to a smooth compact totally real submanifold with boundary been studied before? What kind of inequalities may hold for instance? I have similar questions for the absolute value of a holomorphic function, or perhaps the log of the absolute value of a holomorphic function (defined on a Kähler manifold), also restricted to a compact totally real submanifold with boundary.

Here is what I am hoping for. I am hoping to find a lower bound of the function I am minimizing in terms of the values of the function at the boundary. So I am hoping for some kind of weak minimum principle in this setting. What are the weakest known hypotheses under which the minimum principle holds?

I feel that if instead of being totally real, the compact submanifold was a complex submanifold, then this setting was heavily studied before.

Has the problem of minimizing the real part of a holomorphic function (thus of a pluriharmonic function) on a complex manifold restricted to a real smooth (let us say compact, to simplify) submanifold with boundary been studied before? What kind of inequalities may hold for instance?

Here is what I am hoping for. I am hoping to find a lower bound of the function I am minimizing in terms of the values of the function at the boundary. So I am hoping for some kind of weak minimum principle in this setting, under some hypotheses of course.

Here is an idea, but I am not sure if it will work. Convert the constrained optimization problem to an unconstrained one, for instance using Lagrange multipliers, and then try to apply a weak minimum principle. I will have to think more about it. One issue is that the submanifold is real, and not complex, so I will have to think under what conditions my approach will work. In case such ideas ring a bell, and someone has come across an article along these lines, please inform me.