The corresponding minimization problem with the area constraint replaced by a perimeter constraint (whatever notion of perimeter one would consider) is in general not well-posed. Indeed, for every $L>0$, one can construct a sequence of smooth connected and closed sets $K_n\subset\overline\Omega$ of perimeter $L$ approaching a subset of $\partial \Omega$ so that $\lambda_1(\Omega\setminus K_n)\downarrow \lambda_1(\Omega)$ as $n\to\infty$ (notice that by regularity there is no doubt on the notion of perimeter of $K_n$). Therefore, as in Problem 2, we restrict the class of admissible obstacles to convex sets. For a fixed $L\in (0,P(\overline{\Omega}))$, consider the {minimization} problem \begin{equation}\label{prob3} \min \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed and convex, $P(K)=L$}\}. \end{equation} The existence of a minimizer is a consequence of the compactness of the class of convex sets and of the continuity of the perimeter w.r.t Hausdorff convergence of convex sets. Notice that, for particular domains $\Omega$ and small values $L$, it is still possible to have trivial solutions. For example, if the boundary $\partial\Omega$ contains a segment and if $L$ is smaller than twice the length of such a segment, then every segment contained in $\partial \Omega$ of perimeter $L$ is a minimizer. On the other hand, if $L$ is large enough, every minimizer has positive Lebesgue measure, since minimizing sequences will not be able to degenerate to a segment. In any case, one expects that every minimizer touches the boundary $\partial \Omega$. ...

The corresponding maximization problem (of Problem 1) has no solutions. Indeed, one can construct a sequence of closed sets $K_n\subset\overline\Omega$ of Lebesgue measure $A$ so that $\lambda_1(\Omega\setminus K_n)\uparrow \infty$ as $n\to\infty$ (for instance by taking $K_n$ as the union of a given closed set in $\overline \Omega$ of area $A$ with a curve filling $\overline \Omega$ as $n$ increases, see [9], [10] where the limit distribution in $\overline{\Omega}$ of such curves is studied in detail). To guarantee the existence of a maximizer one needs to prevent maximizing sequences to spread out over $\overline\Omega$. This can be achieved by imposing stronger geometrical constraints on the class of admissible obstacles (notice that connectedness is still not sufficient). Therefore, for a fixed $A\in (0, \mathcal{L}(\Omega))$, we are led to consider the maximization problem \begin{equation}\label{prob2} \max \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed and convex, $\mathcal{L}(K)=A$}\}. \end{equation} Now, the existence of a maximizer in the restricted class of convex sets is straightforward (see [3], [5]). Moreover, as convexity seems necessary for the existence, it is natural to expect every solution of this maximization problem to saturate the convexity constraint, in the sense that the boundary of any solution should contain non-strictly convex parts. In particular, it would be interesting to know whether this maximization problem has only polygonal sets as solutions, see [7], [8] for results in this direction for shape optimization problems with convexity constraints. ...

For a fixed $A\in (0, \mathcal{L}(\Omega))$, with $\mathcal{L}$ the Lebesgue measure, consider the minimization problem $$ \min \{ \lambda_1(\Omega\setminus K) : \; \text{$K\subset \overline{\Omega}$, $K$ closed, $\mathcal{L}(K)=A$}\}. $$ This problem is related to the minimization of the first eigenvalue among open sets constrained to lie in a given {box} (and also with a given area), see [5]. Indeed, passing to the complementary set $O=\Omega\setminus K$ this problem becomes equivalent to the minimization of $\lambda_1(O)$ among open sets $O\subseteq \Omega$ of area $\mathcal{L}({\overline{\Omega}})-A$ (in this framework $\Omega$ represents the box). Therefore, from what is known on the minimizers contained into a box, we infer the existence of a solutionfor that minimization problem and some of its qualitative properties. We have to distinguish two cases, depending on the existence of disks of area $\mathcal{L}(\overline{\Omega})-A$ that are contained inside $ \Omega$ (to this aim we introduce the inradius $\rho(\Omega)$ of $\Omega$). ...

Let $\mathbf{x} = (x_1,\ldots,x_n)$, and $P(\mathbf{x}) \in \mathbb{Z}[\mathbf{x}]$. A Diophantine equation $P(\mathbf{x}) = 0$ is partition regular if, for any $r$-colouring $c: \mathbb{N} \to r$, there exists $c$-monochromatic $\mathbf{m} = (m_1,\ldots,m_n) \in \mathbb{N}^n$ such that $P(\mathbf{m}) = 0$. ...

Isaac Newton in his Principia (first book Ch. 12, Theorem XXXI) asserts that spherical shells exert no gravitational force in the cavity of the shell. This result was later extended to (ellipsoidal) homoeoid by P.-S. Laplace, using computation, and soon after by J. Ivory, using a more geometric approach; a homoeoid is the domain bounded by two homothetic ellipsoids with a common center. ...

Most of the above questions and open problems can naturally be stated for unbounded domains and half-spaces. Several authors have considered the scalar case of such problems and the literature is vast. The reader may consult paper [8], and the references therein for these problems. ...

In scalar case, the Bernoulli problem discussed earlier can also be obtained as a limit problem of the singular perturbation; literature are vast for this problem, so none mentioned none forgotten. A similar theory, yet to be developed, seems plausible for the system case, which seems to have much more possibilities and variations than its scalar counterpart. ...