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On weak solutions of boundary value problems for some general differential equations V. P. Burskii Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region
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     § 1. IntroductionThe foundations of the general theory of boundary value problems for differential equations without type were laid down by Vishik in [1], in which a boundary value problem was considered as a specification of the domain of some extension of the minimal operator. Later, Hörmander [2] refined the concept of a boundary value problem; he also proved Vishik’s conditions for existence of a well-posed boundary value problem for the case of a scalar differential operation with constant coefficients. At the same time, Lopatinskiĭ[3] obtained a condition for a general differential boundary value problem for an elliptic equation or a system to be of Fredholm type. The surge of common interest of 1960s in this field with subsequent recognition of the difficulties associated with the lack of serious advancements in research was followed by a long decline in interest. Among the studies of 1960s, we mention those by Agranovich [4], Berezanskii (see [5], Chap. II, §§ 2–3), and the later studies by Dezin [6] on smoothly generated general boundary value problems. At present, the study of general settings of boundary value problems takes place in the directions set by Petrovskii [7] in the mid-1960s, where the settings of boundary value problems were linked with a specific type of differential equation, and, among such settings, the relevant task is to single out the correct settings of boundary value problems (see, for example, the books by Bitsadze [8] and Soldatov [9]). Various generalized settings of boundary value problems, and, in particular, the theory of generalized (also called weak) solutions of boundary value problems based on the theory of rigged spaces had also been proposed. Such problems are studied for equations in which the elliptic part is given by divergent-free terms, that is, the development here takes place along the direction outlined already by S. L. Sobolev and O. A. Ladyzhenskaya (see, for example, [10], Chap. II, §§ 2–5). The ideology of the properly elliptical case had accustomed us to the fact that the results of well-posedness of boundary value problems hold for a wide class of domains, and, generally, only boundedness and smoothness of the boundary are indicated in the statements. A similar situation also occurs in general settings of boundary value problems, for example, for divergent linear and quasilinear elliptic equations. The wish to consider the most general equations and systems with such properties has led the author of the present paper to generalized settings of Dirichlet problem, Neumann problem, and other boundary value problems for equations and systems of the form with general (matrix, in general) differential operation $\mathcal L$ and some linear or non-linear operator $A $ on an $L_2(\Omega)$-space. Boundary value problems for equations of the form (1.1) were studied by the author in [11]. The present paper is concerned with problems for equation (1.2).Here, by a weak solution of, for example, a homogeneous Dirichlet problem one means a function from the domain of the minimal operator $u\in D(L_0)$ satisfying the “integral” identity
$$
\begin{equation}
\langle\mathcal{L}u,A\mathcal{L}\phi\rangle_\Omega=\langle f,\phi\rangle_\Omega
\end{equation}
\tag{1.3}
$$
for any $\phi\in C_0^\infty(\Omega)$. The analogy with divergent equations becomes transparent on noting that, for the operator $\mathcal{L}=\nabla$, where $\mathcal{L}^+=-\operatorname{div}$, the domain of the minimal extension $D(L_0)$ coincides with the Sobolev space $H^1_0(\Omega)$, and since identity (1.3) involving, say, the Nemytskii operator $A$, becomes the well-known definition of a generalized (weak) solution of the Dirichlet problem for quasilinear second-order elliptic equation. The methods used by the author in the present paper are from the general theory of boundary value problems. Corresponding examples are given, of which one pertains to an equation with the wave operator $\Box (a(x,\Box u))=f$, for which the Dirichlet problem in a bounded domain is well-posed. The present paper continues the author’s studies in [12]–[15]. § 2. Constructions of the general theory of boundary value problemsLet $\mathcal L = \sum_{|\alpha |\leqslant l} a_\alpha (x) D^\alpha$, where $a_\alpha \in C^{\infty,k\times j }(\overline\Omega)$, $D^\alpha=(-i\partial)^{|\alpha|}/\partial x^\alpha$, is a differential operation of general form with $(k\times j)$-matrices $a_\alpha$, whose elements are smooth complex-valued functions, and let $\Omega$ be an arbitrary bounded domain in $\mathbb{R}^n$. The operation $\mathcal L$ formally generates the adjoint operation $\mathcal L^+ = \sum_{|\alpha |\leqslant l} D^\alpha (a^*_\alpha (x)\,{\cdot}\,)$, where $a^*_\alpha (x)$ is the adjoint matrix. We will consider the following Sobolev spaces: $H=L_2^j(\Omega)$, $H^+=L_2^k(\Omega)$, $H^l=(H^l(\Omega))^j$, $H^l_0=(H_0^l(\Omega))^j$, $H^{+l}=(H^l(\Omega))^k$ and $H^{+l}_0=(H_0^l(\Omega))^k$. The minimal operator $L_0$ defined as the closure of the operator $\mathcal L$ (which is originally defined on $(C_0^\infty(\Omega))^j$) with respect to the norm of the graph $\| u\|^2_L=\| u\|^2_{L_2^j(\Omega)}+ \| \mathcal L u\|^2_{L_2^k(\Omega)}$ and the similar minimal operator $L_0^+$ generate the maximal operators $L=(L_0^+)^*$, $L^+=L_0^*$ via the conjugation operation in Hilbert spaces. The domains $D(L_0)$, $D(L^+_0)$, $D(L)$, $D(L^+)$ of these operators are Hilbert spaces with respect to the corresponding norm of the graph. Let $C(L)=D(L)/D(L_0)$ be boundary space for the operator $L$, and $\Gamma\colon D(L)\to C(L)$ be the quotient mapping; the corresponding definitions of $C(L^+)$ and $\Gamma^+$ for the operator $L^+$ are similar. For the maximal operator $L$, we have the short exact sequence (see [16], Chap. 1, § 2) There is a similar sequence for the minimal operator, and there are also exact quotient sequences $\operatorname{Im}L/\operatorname{Im}L_0$ and $C(L)= D(L)/D(L_0)$. Combining, we get the diagram where the operators $i_C$ and $L_C$ are defined by $i_C(u+\ker L_0)=u+D(L_0)$, $L_C(u+ D(L_0))=Lu+\operatorname{Im}(L_0)$, and the operator $\Gamma_{\ker}$: $\ker L \to C(\ker L):= \ker L/\ker L_0$ is defined by the quotient mapping.The commutativity of all squares is clear. So, diagram (D1) is commutative, and all columns and the two top rows are exact. That the bottom row is exact follows from the algebraic $(3\times3)$-lemma (see [16], Chap. V, § 2). Thus, we have established the following result. Proposition 1. Diagram (D1) is commutative, its rows and columns are exact. Diagram (D1) means, clearly, that the maximal operator $L=L_0\oplus L_{C}$ decomposes as a direct sum. Note that all the constructions and diagram (D1) can also be given for a central Banach space in place of $L_2(\Omega)$; the constraint here is that the Vishik conditions are not justified in this setting (see below). Consider the Vishik conditions (see [1]):
$$
\begin{equation}
-\text{ the operator }L_0\colon D(L_0)\to H^+\text{ has continuous left inverse};
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
-\text{ the operator }L_0^+\colon D(L_0^+)\to H \text{ has continuous left inverse}.
\end{equation}
\tag{2.2}
$$
Standard arguments of functional analysis show that these conditions are equivalent, respectively, to the conditions
$$
\begin{equation}
-\text{ the operator }L\colon D(L) \to H^+ \text{ is surjective};
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
-\text{ the operator }L^+\colon D(L^+) \to H \text{ is surjective}.
\end{equation}
\tag{2.4}
$$
Example 1. Some classes of differential operators satisfying (2.1) and (2.2) in a bounded domain were indicated in [13]. This list includes: i) scalar operators with constant coefficients, ii) scalar operators of principal type, iii) scalar operators of constant strength, iv) matrix operators with constant complex coefficients with the Paneah–Fuglede property, v) matrix operators which are uniformly elliptic in the sense of Douglis–Nirenberg in a domain with smooth b1oundary. Condition (2.1) implies that $\ker L_0=0$, and (2.2) ensures that $\operatorname{Im}L= H^+$. In addition, it is known that if $\operatorname{Im}L_0$ is closed, then there is a decomposition $H^+=\operatorname{Im}L_0\oplus \ker L^+$. So, under conditions (2.1), (2.2), diagram (D1) becomes diagram (D2): One can similarly construct the diagram for the operators $L^+_0$, $L^+$ with projections $\Gamma^+$ and $\Gamma^+_{\mathrm{Im}}$.A homogeneous boundary value problem (see [2], p. 172) is the problem of finding a solution to where $B$ is the subspace in the boundary space $C(L)=D(L)/D(L_0)$ which defines the boundary value problem. Problem (2.5) is said to be well-posed if the operator $L_B=L|_{D(L_B)}$, $D(L_B)= \Gamma^{-1} B$ is a solvable extension of the operator $L_0$, that is, if the operator $L_B\colon D(L_B)\to H^+$ has a continuous inverse (which is also a right inverse of $L$). The following result is well known (Visihik [1]; for Hörmander’s interpretation of this result, see [2], § 3).Proposition 2. The operator $L_0$ has a solvable extension (and the operator $L$ has a well-posed boundary value problem (2.5)) if and only if both conditions (2.1) and (2.2) hols. Thus, Example 1 provides the known classes of operators for which there exists a well-posed boundary value problem in any bounded domain. In the next result, we give an example of a well-posed boundary value problem for a non-elliptic equation in a bounded domain. Proposition 3 (see [13]). The following boundary value problem in the unit disc $D_{0,1}=\{x\in {\mathbb{R}^2}\mid |x|<1\}$ is well-posed in $L_2(D_{0,1})$: here $\Gamma_1=\{|x|=1,\, \pi/2\leqslant \tau \leqslant 2\pi \}$, $\Gamma_2=\{|x|=1,\, \pi \leqslant \tau \leqslant 3\pi/2\}$, and $\tau$ is the angular variable. The adjoint problem to problem (2.5) is the boundary value problem where the space
$$
\begin{equation*}
B^+=\Gamma^+D^+_B,\qquad D^+_B=\{v\in D(L^+)\mid \forall\, u\in\Gamma^{-1}(B),\ [u,v]=0\}
\end{equation*}
\notag
$$
is generated by the Green form $[u,v]=\int_\Omega(Lu \cdot \overline v-u \cdot \overline{L^+v})\, dx$.
§ 3. Generally posed boundary value problemsBelow, we will use the notation from the theory of rigged spaces (see [5]). If $H$, $H^+$ are Hilbert spaces and $H\supset H^+$ is a dense embedding with topology, then $H^-=(H^+)^\prime$ is the Hilbert space which is dual to $H^+$ relative to the norm of $H$; this space will be called the dual space to $H^+$ (relative to the central space $H$). If there are two such spaces $H^+$ and if we are given a continuous linear operator $U\colon H^+_1\to H^+_2$, then the corresponding adjoint operator $U^*=U'\colon H^-_2\to H^-_1$ is called the dual operator to $U$. Below, $H=L_2^{k}(\Omega)$. Now let $A\colon H\to H$ be a linear or non-linear continuous operator. Consider the equation Definition 1. A function $u\in D(L_B)$ satisfying the integral identity for any function $v\in D(L_B)$ will be called a weak solution of the problem $\Gamma u\in B$, $\Gamma^+ALu\in B^+$, generated by the problem $Lu=f$, $\Gamma u\in B$ in the domain $\Omega$ for equation (2.1) with a right-hand side $f$ from $D'(L_B)$ and an operator $A$. Here, the space $B^{+}$ defines the problem $L^+v=g$, $\Gamma^+ v\in B^+$, which is adjoint to the problem $Lu=f_1$, $\Gamma u\in B$ (see (2.7)). The integral identity (3.2) can be written as an equation with the operator $L_B^\prime$ dual to $L_B$: In particular, problem (3.2) will be called the generalized Dirichlet problem if $B= 0$ (that is, $L_B=L_0$); it is called the generalized Neumann problem if $B=C(L)$ (that is, $L_B=L)$. As in the previous section, these definitions are based on an analogy with the classical definitions of the corresponding problems for a quasilinear divergent elliptic equation with the gradient $\nabla$ (for second-order equations) or the generalized gradient $\nabla^2,\dots,\nabla^m$ (for higher-order equations) in place of the operation $\mathcal L$, and the Nemytskii operator in the role of $A$. Definition 2. A generalized boundary value problem (3.2) will be said to be well-posed if the operator $\widetilde{L}_{BA}=L_B'\cdot A\cdot L_B\colon D(L_B)\to D'(L_B)$ has a two-sided inverse $M\colon D'(L_B)\to D(L_B)$; it is called normally well-posed if, for any function $f\in D'(L_B)$ orthogonal to the kernel $\ker L_B$, there exists a unique (up to an additive component $h\in \ker L_B$, function $u\in D(L_B)$ satisfying equation (3.3) and depending continuously on $f$. This function $u\in D(L_B)$ is known as a weak solution of this problem. The following result holds. Proposition 4. Problem (3.2) with $L_2^{k}(\Omega)$-continuous operator $A$ is normally well-posed if and only if the operator $L_B$ is normally solvable and the operator $P\cdot A$ is a homeomorphism from the closed subspace $\operatorname{Im}L_B$ onto itself. Here, $P\colon L^{k}_2(\Omega)\to \operatorname{Im}L_B$ is the orthogonal projection. Problem (3.2) is well-posed if and only if it is normally solvable and $\ker L_B=0$. Proof. The sufficiency is clear. Let us prove the necessity for well-posedness of problem (3.2). Assume that $\widetilde{L}_{BA}$ is a homomorphism with inverse operator $M$; let $L_B$ be an arbitrary extension. We first note that the equalities $ML_B'AL_Bu=u$ and $L_B'AL_BMv=v$ imply that $L_B$ is injective and $L_B'$ is surjective. The space $\operatorname{Im}L_B$ equipped with the topology induced by the injection $\widetilde{L}_B\colon D(L_B)\to \operatorname{Im}L_B$ from $D(L_B)$ is not in general a closed subspace of $H=L_2^{k}(\Omega)$, but this space is compactly embedded in some closed subspace $H_1\stackrel{J}{\subset} H$, and the embedding $I\colon \operatorname{Im}L_B\subset H_1$ is continuous. The dual operator to $I$ also generates a continuous dense embedding $I'\colon H_1\subset \operatorname{Im}'L_B$. Consider the orthogonal projection $P\colon L_2^{k}(\Omega)\to H_1$. We now have a sequence of mappings $\operatorname{Im}L_B\xrightarrow{I}H_1 \xrightarrow{PAJ} H_1 \xrightarrow{I'}\operatorname{Im}'L_B$, whose composition is a homeomorphism with inverse operator $M_1=\widetilde{L}_BM\widetilde{L}_B'$. For any function $u\in I\operatorname{Im}L_B\subset H_1$, we have the equality $IM_1I'PAJu=u$, and hence, since the operators under consideration are continuous, we approximate an arbitrary function $w\in H_1$ by a sequence $w_n\to w,w_n\in \operatorname{Im}L_B$. Making $n\to\infty$ in the last equality, we find that this equality also holds for any $w\in H_1$. Hence $I$ is surjective. So, the operator $L_B$ is normally solvable, and the topology of $\operatorname{Im}L_B$ coincides with that induced from $H$. Finally, since $I$ and $I'$ are isomorphisms, the operator $PA=PAJ$ is a homeomorphism onto the space $\operatorname{Im}L_B$.
In order to be able to deal with the case of normal well-posedness of problem (3.2), it suffices to take the quotients of the spaces $D(L_B)$ and $D'(L_B)$ by $\ker L_B $ and $\bot \ker L_B$, respectively, and use the same arguments. Remark 1. If we are given a Fréchet differentiable non-linear operator $A$, then, in order to secure the existence of a locally continuous (on some $D\subset \operatorname{Im}L_B$) inverse operator of $PA$, it suffices to verify that there exists an inverse operator of the Fréchet derivative $PA'(u)$ of the operator $PA$ at any point $u\in D$, and then apply the inverse function theorem. In this case, the condition $\| PA'(u)h\| \ \geqslant \ C\| h\|$ for the existence of a bounded inverse of $PA'(u)$ can be written as the estimate or as the inequality $\exists\, C>0$, $\forall\, u$, $\forall\, h$, $\exists\, w$, $\langle A'(u)h,w\rangle \geqslant C\|h\| \, \|w\|$. The last estimate holds if there exists an operator $V\colon D\times \operatorname{Im}L_B\to \operatorname{Im}L_B$ such that $\langle A'(u)h,V(u,h)\rangle \geqslant C\|h\| \, \|V(u,h)\|$. In the simplest case of a continuous linear operator $V(u,h)=Vh$, we get a sufficient condition for local homeomorphicity of the operator $PA$ to the effect that the composition of some operator $V^*$ and the Fréchet derivative $A'(u)$ should be positive definite on $\operatorname{Im}L_B$: The next result follows from Proposition 4. Proposition 5. Let $A\colon H\to H$ be a continuous operator and the extension $L_B$ be solvable. Then 1) problem (3.2) is well-posed if and only if $A$ is a homeomorphism, 2) if the operator $A$ is surjective, then problem (3.2) is solvable, that is, it has a weak solution, Example 2. As an operator $\mathcal{L}$ we consider an arbitrary scalar differential operator with constant coefficients, and as $A$ we take the Urysohn operator where for all $x,t\in \Omega$ and all $\xi_1,\xi_2\in \mathbb{R}$, § 4. Operators with subordinate termsThe above arguments cannot be applied to operators involving lower-order derivatives. For example, Example 2 does not cover equations with $K=K(x,t,u(t)$, $ \nabla u(t),\Delta u(t))$. Below, we will present an approach capable of dealing with lower-order terms. Another possibility for this comes from vector operations of type $\mathcal{M}=(\mathcal{L},1)$, bearing in mind that in this case $\mathcal{M}^+(v_1,v_2)=\mathcal{L}v_1+ v_2$, $\mathcal{M}^+\mathcal{M} u=\mathcal{L}^+\mathcal{L}u+u$. Now, the operator $A$ is allowed to involve lower-order terms, which enter compactly into $A$. Consider the operator $\widetilde{L}_{BA}$ acting by $\widetilde{L}_{BA}u=$ $L_B'A(u,L_Bu)=L_B'\widetilde{A}(\mathcal{K}u,L_Bu)$, where $\mathcal{K}\colon D(L_B)\to L_2^m(\Omega)$ is some compact operator, the extension $L_B$ is normally solvable, $\widetilde{A}\colon L_2^m(\Omega)\times \operatorname{Im}L\to \operatorname{Im}L$ is a continuous operator such that $PA(u,w)=\widetilde{A}(\mathcal{K}u,w)$, where $A\colon D(L_B)\times \operatorname{Im}L\to L_2^{k}(\Omega)$ and the orthogonal projection $P\colon L_2^{k}(\Omega)\to \operatorname{Im}L$ is as above. We will consider the following conditions:
$$
\begin{equation}
-\ \forall\, v\in L_2^m(\Omega) \text{ the operator }\widetilde{A}(v,{\cdot}\, )\colon \operatorname{Im}L\to \operatorname{Im}L\text{ is a homeomorphism},
\end{equation}
\tag{4.1}
$$
$$
\begin{equation}
-\text{ the homeomorphism }(\widetilde{A}(v,{\cdot}\,))^{-1}\colon \operatorname{Im} L\to \operatorname{Im}L\text{ is uniformly bounded},
\end{equation}
\tag{4.2}
$$
that is, there exists a monotone function $\beta \colon \mathbb{R}^{+}\to \mathbb{R}^{+}$, $\beta(r)=R$ such that the range $(\widetilde{A}(v,{\cdot}\,))^{-1}(S(0,r))$ of any ball $S(0,r)$ $\subset \operatorname{Im}L$ of radius $r$ lies in the ball $S(0,R)\subset \operatorname{Im}L$ for each $v\in L_2^m(\Omega)$. Note that in order to verify condition (4.2) it suffices to show that the mapping
$$
\begin{equation}
(\widetilde{A}(v,{\cdot}\,))^{-1}\colon L_2^{k}(\Omega)\to L^{k}_2(\Omega)
\end{equation}
\tag{4.3}
$$
is uniformly bounded if the operator $\widetilde{A}(v,\cdot )\colon L_2^{k}(\Omega)\to L_2^{k}(\Omega)$ is defined and invertible on the entire $L_2^{k}(\Omega)$. A function $u\in D(L_B)$ satisfying the integral identity for any function $v\in D(L_B)$ will be called a weak solution of the problem $\Gamma u\in B$, $\Gamma^{+}A(u,Lu)\in B^{+}$, generated by the problem $\Gamma u\in B$ in the domain $\Omega$ for the equation with an arbitrary function $f\in D'(L_B)$. The integral identity (4.4) corresponds to the equation
$$
\begin{equation}
\widetilde{L}_{BA}u=L_B'\cdot\widetilde{A}(\mathcal{K}u,L_B\,u)=f.
\end{equation}
\tag{4.6}
$$
A generalized boundary value problem (4.4) will be called solvable if, for any right-hand side $f\in D'(L_B)$, there exists a function $u\in D(L_B)$ obeying (4.6). A generalized boundary value problem (4.6) will be called well-posed if the operator $\widetilde{L}_{BA}\colon D(L_B)\to D'(L_B)$ has a two-sided continuous inverse $M\colon D'(L_B)\to D(L_B)$. From these definitions, we have the following results. Proposition 6. Assume that the extension $L_B$ is normally solvable and $\ker L_B= 0$. Then 1) the generalized problem (4.4) is solvable (well-posed, respectively) if and only if the equation has a solution for any function $f\in \operatorname{Im}L_B$ (is well-posed for such an $f$, that is, the operator of this equation has a continuous inverse);2) if conditions (4.1), (4.2) are met, then the generalized problem (4.4) is solvable. Proof. Assertion 1) follows (as in the proof of Proposition 4) from the fact that $L_B\colon D(L_B)\to L_2^{k}(\Omega)$ is an isomorphism onto its range, and since $\ker L_B'\bot \operatorname{Im}L_B$.
2) Let $f\in D'(L_B)$ be an arbitrary function. From Proposition 5 and condition (4.1) it follows that the mapping
$$
\begin{equation*}
T\colon D(L_B)\ni u\quad \to\quad L_B^{-1}(PA(u,\cdot ))^{-1}((L_B')^{-1}f)\in D(L_B)
\end{equation*}
\notag
$$
is a completely continuous operator. By condition (4.2), for each ball $F\ni f$, there exists a ball $U\subset \operatorname{Im}L_B$ which lies in the pre-image $(\widetilde{A}(\mathcal{K}u,{\cdot}\,)^{-1}((L_B')^{-1}F)$ for all $u$. Next, the compact mapping $L_BTL_B^{-1}$ carries the closure $\overline{U}$ of the ball $U$ into itself. Hence $L_BTL_B^{-1}$ has a fixed point by Schauder’s fixed point theorem. Therefore, problem (4.4) is solvable. Remark 2. We would like to be able to replace $\mathcal{K}u$ by a family of some differential expressions, but in this case we should require that the operators generated by these differential expressions be compact. This suggests the following definition. A differential operation $\mathcal{M}$ will be called $B$-subordinate to an operation $\mathcal{L}$ (written $\mathcal{M}\,{\prec \prec_B}\, \mathcal{L}$) if $D(M)\supset D(L_B)$ and if the operator $I\circ M|_{D(L_B)}\colon D(L_B)\to L_2(\Omega)$ with embedding $I\colon \operatorname{Im} M|_{D(L_B)}\to L_2(\Omega)$ is compact. Here the embedding is dense, which means that the a priori estimate In [2], conditions on the symbols of operators were given under which such comparisons hold. Of course, derivation of some conditions for comparison of operators from different classes is an important and difficult problem. Example 3. Consider the equation § 5. Non-linear problems involving the Nemytskii operatorAs an operator $A(\,{\cdot}\,,{\cdot}\,)$ from the previous section, we will consider the Nemytskii operator generated by a finite-dimensional mapping $g\colon (Au)(x) = g(x,u(x))$, $g\colon \Omega \times \mathbb{R}^r\to \mathbb{R}^r$, $\Omega \subset \mathbb{R}^n$, or $(AU)(x)=a(x,U(x))$, $a\colon \Omega \times \mathbb{R}^N=\Omega \times \mathbb{R}^M\times \mathbb{R}^r\to \mathbb{R}^r$, $a(x,\eta,\xi)=\{A_\alpha(x,\eta,\xi)\}$ according to whether the equation under consideration is of type (3.1) or (4.5). 5.1. Some known resultsWe first consider the operator $A\colon L_2^r(\Omega)\to L_2^r(\Omega)$ generated by the mapping $g\colon (Au)(x)=g(x,u(x))$, where $g\colon \Omega \times \mathbb{R}^r\to \mathbb{R}^r$ satisfies the Carathéodory condition: $g(x,\xi)$ is continuous with respect to $\xi$ for almost all $x$ and is measurable with respect to $x\in \Omega$ for all $\xi$. It is known that this operator is continuous and bounded in $L_2^r(\Omega)$ if and only if
$$
\begin{equation*}
\exists \,g_1\in L_2(\Omega),\ \exists\, C> 0,\ \forall\, x,\ \forall\, \xi\qquad |g(x,\xi)|\leqslant g_1(x)+C|\xi|.
\end{equation*}
\notag
$$
The following fact is a direct consequence of the above estimate. Proposition 7. Let a mapping $g(x,u)$, $g\colon \Omega \times \mathbb{R}^r\to \mathbb{R}^r$ satisfy the Carathéodory condition. Assume, in addition that, for almost all $x$, $g(x,{\cdot}\, )$ has a continuous inverse $g^{-1}(x,{\cdot}\,)$, $g^{-1}(x,g(x,\xi ))=g(x,g^{-1}(x,\xi))=\xi$, and satisfies the linear growth condition at infinity: Then the Nemytskii operator $A$ corresponding to this mapping is a homeomorphism in the space $L_2^r(\Omega)$, and vice versa. Let us recall some known definitions adopted in the theory of operators of non-linear boundary value problems. An operator $T\colon X\to X'$ from a reflexive space into its dual is said to be (see [17]): i) monotone if $\langle Tu-Tv,u-v\rangle\geqslant 0$ for all $u$, $v$; ii) strictly monotone if $\langle Tu-Tv,u-v\rangle >0$ for all $u\neq v$; iii) radially continuous if, for each $u,v\in X$, the function $s\to \langle T(u+ sv),v\rangle$ is continuous on $[0,1]$; iv) coercive if $\langle Tu,u\rangle \geqslant \gamma(\|u\|)\|u\|$ for some monotone function $\gamma \colon [0,\infty)\to \mathbb{R}$, $\lim_{s\to \infty}\gamma(s)=+\infty$; v) satisfying the $S$-property of F. Browder (or the Skrypnik $\alpha$-condition [18]) if $u_n\to u$ in $X$ whenever $\langle Tu_n-Tu,u_n-u\rangle \to 0$ and $u_n\rightharpoonup u$. These properties are necessary for the existence of a continuous inverse operator. In particular, the following facts hold (see [17]). Proposition 8. 1. If condition ii) is satisfied either for $T$ or for $(-T)$, then $T$ is injective. 2. Let $T$ be a radially continuous monotone coercive operator. Then $T$ is surjective, and for all $f\in X'$, the set $T^{-1}(f)$ is weakly convex and closed (the Browder–Minty theorem). 3. If $T$ is a radially continuous monotone coercive operator with $S$-property, then $T^{-1}$ is continuous. For equations of the form $\mathcal{A}u\equiv \sum_{|\alpha |\leqslant m}D^\alpha [A_\alpha (x,u,\nabla u,\dots,\nabla^mu)-f_\alpha]=0$ (that is, for equation (4.5) with operator $A$ generated by the mapping $a\colon AU=a(x,U(x))$, $a\colon \Omega \times \mathbb{R}^N\to \mathbb{R}^N$, $a(x,\eta,\xi )=\{A_\alpha (x,\eta,\xi)\}$), there are examples of classes of mappings $a$ satisfying properties i)–iv), so that well-posedness of the corresponding boundary value problems can be studied (see [17]–[19]). Properties ii) (and also v) and iv)) show that the equations in such problems should be elliptic. This is one possible approach (for other methods, see, for example, [19]). The following important observation is due to H. Gajewski (see [17]). Proposition 9. Let $X$, $Y$ be Banach spaces. If $L\colon X \to Y$ is a linear isometric isomorphism and an operator $A\colon Y\to Y'$ satisfies some of conditions i)– v), then the operator $L'AL\colon X\to X'$ also satisfies this condition. Note that below the requirement that $L$ is isometric can be disposed of, because in the study of continuity problems we can always change to an equivalent metric. Following [17], consider the operator $A$ defined by
$$
\begin{equation}
\begin{aligned} \, (Au)(x) &=\{a_1(x,u(x)),\dots,a_r(x,u(x))\}, \\ a_i(x,u) &=\varphi(x,|u|)\sum_{j=1}^rb_{i,j}(x)u_j. \end{aligned}
\end{equation}
\tag{5.2}
$$
Proposition 10. Let the function $\varphi $ in (5.2) be bounded and satisfy the Carathéodory condition. Next, let $(b_{i,j}(x))$ be a symmetric positive definite real matrix consisting of bounded functions. Let, in addition, the function $\varphi $ satisfy the estimate $\varphi(x,t)t-\varphi (x,s)s\geqslant m(t-s)$ for $t\geqslant s,m>0$. Then assumptions ii)– v) are met, and hence, the corresponding operator $A$ is a homeomorphism in $L_2^r(\Omega)$. 5.2. Applications to problems with surjective extensionIn this section, we will show which conclusions can be made from the analysis of the previous section for the Nemytskii operator $A$. Unfortunately, in the majority of cases, the Nemytskii operator is Fréchet differentiable only on a dense subset of $L_2(\Omega)$, and so, in this case, the approach of Remark 3 cannot be used. However, by employing Proposition 7, this difficulty can be circumvented either by constructing an explicit expression for the inverse of $PA$, or by using the monotonicity of the operator $PA$ (or, possibly, of $PMAM^*$ with some invertible $M$). We first study the most simple case where the operator $L_B$ is solvable, that is, $\operatorname{Im}L_B=L_2(\Omega)$, $P=\mathrm{id}$, $\ker L_B=0$. Let $a=g$ depend only on $x$ and $\xi$. Consider the generalized boundary value problem (3.2) for the quasilinear equation (3.1), where $A\colon L_2^r(\Omega)\to L_2^r(\Omega)$ is the Nemytskii operator satisfying the assumption of Proposition 7. Having at our disposal Proposition 8, we can speak about well-posedness of this problem if the operator $L_B$ is invertible, and vice versa. Similar arguments can also be applied if the homeomorphism $A$ is given by (5.2) in which the functions $a_j$ satisfy the hypotheses of Proposition 10. The following result holds. Proposition 11. Let an operator $A\colon L_2^r(\Omega)\to L_2^r(\Omega)$ satisfy either the conditions of Proposition 7 or the conditions of Proposition 10. Then 1) if the extension $L_B$ is solvable, then the generalized problem (3.2) is well-posed; 2) if the condition (2.3) for surjectivity of the operator $L$ is met, then the generalized Neumann problem (3.2), $B=C(L)$, is normally solvable; in particular, this problem is well-posed for equation (3.1) in which the operator $\mathcal{L}$ is any operator from Example 1. Example 4. From Proposition 3 and the above remarks, one can easily verify well-posedness of the following non-linear problem in the unit disc $D_{0,1}=\{x\in \mathbb{R}^2\mid |x|<1\}$ generated by problem (2.6): Now let a mapping $a$ be such that $a\colon \Omega \times \mathbb{R}^M\times \mathbb{R}^r\to \mathbb{R}^r$, $\Omega \subset \mathbb{R}^n$, $a(x,\eta,\xi)=\{A_\alpha(x,\eta,\xi),\, |\alpha|\leqslant m\}$. In this case, we will use the approach proposed in Proposition 9. In order that the operator $A$ satisfy conditions (4.1), (4.2), the mapping $a(x,\eta,{\cdot}\,)$ should obey the conditions of Proposition 7 or in Proposition 10 uniformly with respect to $\eta$, that is, the function $g_1\in L_2(\Omega)$ and the constant $C_1$ from Proposition 7 or the constant $m>0$ and the positive definiteness constant of the matrix $(b_{i,j}(x))$ should not depend on $\eta$. As above, condition (4.1) is met. Property (4.3) holds by constructions of the inverse operator in Propositions 7 and 10. Using assertion 2) of Proposition 11, we arrive at the following results. Proposition 12. Consider the equation where $L_j\prec \prec_BL$ (this means that the composition $I_j\circ L_j\colon D(L_B)\to L_2(\Omega)$ with the embedding operator $I_j$ is compact). Assume that the mapping $a(x,\eta,{\cdot}\,)$ satisfies the conditions in Propositions 7 or 10 uniformly with respect to $\eta$. Then 1) if the extension $L_B$ is solvable, then so is problem (4.4) generated by this operator; 2) if condition (2.2) is met, then the Neumann problem (4.4), $B=C(L)$, corresponding to the operator $A(u,L_Bu)$ is solvable for $f\in D'(L)$, $f\perp \ker L$; in particular, this problem is solvable for equation (4.5) in which the operator $\mathcal{L}$ is one of the operators in Example 1. Example 5. From Proposition 12 in view of Proposition 3 and Example 4 it follows that the following non-linear problem is solvable: $\langle a(x,u,\Box_B\,u),\Box_B\,v\rangle=\langle f, v\rangle$, where the function $a(x,\eta,\xi )$ satisfies the hypotheses of Proposition 7 of 10 uniformly with respect to $\eta$, because $D(\Box_B)\subset H^1(\Omega)\subset \subset L_2(\Omega)$. 5.3. Applications to problems with non-surjective extensionAssume that the range $\operatorname{Im}L_B$ does not coincide with the whole space $L_2(\Omega)$. In order to be able to employ Propositions 7, 10, it is necessary that the operator $PA$ be continuously invertible on $\operatorname{Im}L_B$. In the general case, the approach of Proposition 7 cannot be applied to guarantee this claim for each subspace $\operatorname{Im}L_B$. To circumvent this issue, we will use Proposition 8, which provides a possibility for that. The following result is a corollary to Propositions 8 and 9. Proposition 13. Let $H$ be a Hilbert space, $G$ be a closed subspace of $H$, $I\colon G\to H$ be an embedding operator, and $P\colon H\to G$ be the orthogonal projection. Next, let an operator $A\colon H\to H$ satisfy at least one of conditions i)– v). Then the operator $PAI\colon G\to G$ also obeys this property. In addition, If the operator $A$ satisfies all these conditions, then the operator $PA=PAI\colon G\to G$ is a homeomorphism for each closed $G\subset H$. For a proof, it suffices to assume that $H$ is a central rigged space (see [16]), take $L=I$ in Proposition 9, with identification $G'=G$, and then apply Proposition 8. Now we can employ the approach of Proposition 7 and 9. Proposition 14. Let an operator $A\colon L_2^r(\Omega)\to L_2^r(\Omega)$ defined by (5.2) satisfy the hypotheses of Proposition 10. Then 1) if the extension $L_B$ is normally solvable, then the generalized problem (3.2) is normally well-posed; if, in addition, $\ker L_B=0$, then the problem is well-posed; 2) the generalized Dirichlet problem (3.2), $B=\{0\}$, is normally well-posed if the operator $L$ is normal, that is, the subspace $\operatorname{Im}L$ is closed in $H^+$; this problem is well-posed if condition (2.1) is met. In particular, the Dirichlet problem for equation (3.1) in which $\mathcal{L}$ is one of the operators in Example 1 is well-posed. Example 6. Consider the well-known generalized statement for the Dirichlet problem for the quasilinear equation in which $A\colon L_2^n(\Omega)\to L_2^n(\Omega)$ is some continuous mapping satisfying conditions ii)–v). As such a mapping one can consider, for example, the Nemytskii operator defined via a mapping $a$ acting in $\mathbb{R}^n$ (see Proposition 10). This problem is well posed by Proposition 14 in view of the Friedrichs inequality. Example 7. Consider another example of the Dirichlet problem in a planar domain for the quasilinear equation with the wave operator $\Box =\partial^2/(\partial x_1\partial x_2)$ and the operator $A\colon L_2(\Omega)\to L_2(\Omega)$ defined via the mapping $a$ in $\mathbb{R}^1$ (see Proposition 10). By Proposition 14, this problem is well-posed under the assumptions of Proposition 10. If, in addition, $a$ depends on $u$, then this problem is solvable. Remark 3. In the present paper, we have used one set of assumptions (of many possible) about an operator $A$ under which the equation $Av=g$ is well-posed (or solvable) in the space $L_2(\Omega)$. This, in turn, implies that the equation $L_B'AL_Bu=f$ is well-posed (or solvable) in the space $D(L_B)$. However, there are other sets of assumptions (for example, boundedness, semicontinuity, coercivity, and pseudomonotonicity of the operator $A$). Each of these conditions can be used in derivation of results similar to Proposition 14. 
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